46 2 Physical and Technical Background Side view Light sheet 100 mm 200 mm 60 mm Top view Thickness Fig. 2.28. Light sheet optics 100 mm 200 mm 60 mm using three cylindrical lenses. Furthermore cases c and d in figure 2.29 should be used in order to min- imize aberrations. For other situations it might be possible to tilt the lens slightly in order to avoid reflections on to other lenses or towards the laser or even into the resonator. 2.5 Volume Illumination of the Flow Whereas most conventional PIV investigations utilize light sheet illumination, they are typically not a practical source of illumination for micro-flows, due to a lack of optical access along with significant diffraction in light sheet forming optics. Consequently, the flow must be volume illuminated, leaving two choices for the visualization of the seed particles – with an optical system whose depth of field exceeds the depth of the flow being measured or with an optical system whose depth of field is small compared to that of the flow. Both of these techniques have been used in various implementations of μPIV. Cummings [281] uses a large depth of field imaging system to explore electrokinetic and pressure-driven flows. The advantage of the large depth of field optical system is that all particles in the field of view of the optical system are well-focused Not recommended Recommended a) c) d) b) Fig. 2.29. General consider- ations on the orientation of lenses inside the light sheet optics. 2.5 Volume Illumination of the Flow 47 and contribute to the correlation function comparably. The disadvantage of this scheme is that all knowledge of the depth of each particle is lost, resulting velocity fields that are completely depth-averaged. For example in a pressure- driven flow where the velocity profile is expected to be parabolic with depth, the fast moving particles near the center of the channel will be focused at the same time as the slow moving particles near the wall. The measured velocity will be a weighted-average of the velocities of all the particles imaged. Cummings [281] addresses this problem with advanced processing techniques that will not be covered here. The second choice of imaging systems is one whose depth of field is smaller than that of the flow domain, as shown in Figure 2.30. The optical system will then sharply focus those particles that are within the depth of field δ of the imaging system while the remaining particles will be unfocused – to greater or lesser degrees – and contribute to the background noise level. Since the optical system is being used to define thickness of the measurement domain, it is important to characterize exactly how thick the depth of field, or more appropriately, the depth of correlation zcorr, is. The distinction between depth of field and depth of correlation is an important although subtle one. The depth of field refers to distance a point source of light may be displaced from the focal plane and still produce an acceptably focused image whereas the depth of correlation refers to how far from the focal plane a particle will contribute significantly to the correlation function. The depth of correlation can be calculated starting from the basic principles of how small particles are imaged [300, 408]. Fig. 2.30. Schematic showing the geometry for volume illumina- tion particle image velocimetry. The par- ticles carried by the flow are illuminated by light coming out of the objective lens (i.e. upward). 48 2 Physical and Technical Background 2.6 Imaging of Small Particles 2.6.1 Diffraction Limited Imaging This section provides a description of diffraction limited imaging, which is an effect of practical significance in optical instrumentation, and of particular interest for PIV recording. In the following we will restrict our description of imaging by considering only one-dimensional functions. If plane light waves impinge on an opaque screen containing a circular aperture they generate a far-field diffraction pattern on a distant observing screen. By using a lens – for example an objective in a camera – the far field pattern can be imaged on an image sensor close to the aperture without changes. However, the image of a distant point source (e.g. a small scattering particle inside the light sheet) does not appear as a point in the image plane but forms a Fraunhofer diffraction pattern even if it is imaged by a perfectly aberration-free lens [11]. A circular pattern, which is known as the Airy disk, will be obtained for a low exposure. Surrounding Airy rings can be observed for a very high exposure. Using an approximation (the Fraunhofer approximation) for the far field it can be shown that the intensity of the Airy pattern represents the Fourier transform of the aperture’s transmissivity distribution [10, 18]. Taking into account the scaling theorem of the Fourier transform, it becomes clear that large aperture diameters correspond to small Airy disks and small apertures to large disks as can be seen in figure 2.31. The Airy function is equivalent to the square of the first order Bessel function. Therefore, the first dark ring, which defines the extension of the Airy disk, corresponds to the first zero of the first order Bessel function shown in figure 2.32. The Airy function represents the impulse response – the so-called point spread function – of an aberration-free lens. We will now determine the diameter of the Airy disk ddiff, because it represents the smallest particle image that can be obtained for a given imaging configuration. In figure 2.32 the value of the radius of the ring and therefore of the Airy disk can be found for a given aperture diameter Da and wavelength λ: Fig. 2.31. Airy pat- terns for a small (left hand side) and a larger aperture diameter (right hand side). 2.6 Imaging of Small Particles 49 1.0 0.8 Airy function Gaussian approximation 0.6 I/ Imax 0.4 0.2 Fig. 2.32. Normalized intensity distribution 0.0 of the Airy pattern and 0.0 0.5 1.0 1.5 2.0 2.5 its approximation by a x/x0 Gaussian bell curve. I(x) d =0 ⇒ diff =1.22 Imax 2x0 with λ x0 = . Da If we consider imaging of objects in air – the same media on both sides of the imaging lens – the focus criterion is given by (see figure 2.33): 1 1 1 + = (2.6) z0 Z0 f where z0 is the distance between the image plane and lens and Z0 the distance between the lens and the object plane. Together with the definition of the magnification factor z M = 0 Z0 the following formula for the diffraction limited minimum image diameter can be obtained: ddiff =2.44 f#(M +1)λ (2.7) where f# is the f-number, defined as the ratio between the focal length f and the aperture diameter Da [10]. In PIV, this minimum image diameter ddiff will only be obtained when recording small particles – of the order of a few microns – at small magnifications. For larger particles and/or larger magnifications, the influence of geometric imaging becomes more and more dominant. The image of a finite-diameter particle is given by the convolution of the point spread function with the geometric image of the particle. If lens aberrations can be neglected and the point spread function can be approximated by the 50 2 Physical and Technical Background Y Image plane Object Z f f plane y Fig. 2.33. Geometric Z Z 0 0 image reconstruction. Airy function, the following formula can be used for an estimate of the particle image diameter [53]: 2 2 dτ = (Mdp) + ddiff . (2.8) This expression is dominated by diffraction effects and reaches a constant value of ddiff when the size of the particle’s geometric image Mdp is con- siderably smaller than ddiff. It is dominated by the geometric image size for geometric image sizes considerably larger than ddiff where dτ ≈ Mdp In practice the point spread function is often approximated by a normal- ized Gaussian curve also shown in figure 2.32 and defined by: 2 I(x) − x =exp 2 (2.9) Imax 2σ √ where the parameter σ must be set to σ = f#(1 + M)λ 2 /π,inorder to approximate diffraction limited imaging. This approximation is particu- larly useful because it allows a considerable simplification of the mathematics encountered in the derivation of modulation transfer functions, which also includes other kinds of optical aberrations of the imaging lens as will be described later. In practice there are two good reasons for optimizing the particle image diameter: First, an analysis of PIV evaluation shows that the error in velocity measure- ments strongly depends on the particle image diameter (see e.g. section 5.5.2). For most practical situations, the error is minimized by minimiz- ing both the image diameter dτ and the uncertainty in locating the image centroid or correlation peak centroid respectively. Second, sharp and small particle images are particularly essential in order to obtain a high particle image intensity Imax, since at constant light energy scattered by the tracer particle the light energy per unit area increases ∼ 2 quadratically with decreasing image areas (Imax 1/dτ ). This fact also explains why increasing the particle diameter not always compensates for insufficient laser power. 2.6 Imaging of Small Particles 51 Equation (2.8) shows that for a range of particle diameters greater than the wavelength of the scattered light (dτ λ), the diffraction limit becomes less important and the image diameter increases nearly linearly with increasing particle diameter.