ratio accompanied by a lower correlation coefficient. To solve this problem, the time sepa-ration method can be applied as long as the sepasepa-ration between the first orthogonal polarised light pulse pair is negligible, e.g. the position of the particle image remains constant in terms of digital registration. In this case, which holds for a wide range of flow velocities, a difference between instantaneous and time separated measurements cannot be observed. Technologi-cally, this can be done by transferring the first image right behind the first illumination.
4.7 Monochromatic aberrations
Before the light scattered by the particles can be recorded it interacts partially with optical elements like glass-window, mirror, beam-splitter and various lenses in order to alter the ori-entation of the field, to separate the incoming light according to the state of polarisation and to generate an image of the tracer particles. Beside a possible loss of light due to absorption or reflection, these components introduce aberrations of different type, direction and magni-tude which may limit the measurement accuracy [8]. Although it is impossible to eliminate all aberrations completely in any real system of finite aperture, a basic understanding of their origin and dependence from optical parameters is the key to reduce these undesirable effects under the resolution of the recording medium or to eliminate certain aberrations completely by accepting aberrations of other types which are of no harm in PIV. The last point is of pri-mary importance because higher order aberrations like distortion1 and curvature of the field just influence the position and form of the image but do not lower the resolution. They can be completely eliminated by calculations according to section 2.4 and do not need to be consid-ered here. Primary aberrations on the other hand like spherical aberration, coma and especially astigmatism deteriorate the image and alter the shape in a characteristic way. This leads to an increased measurement error as the performance of the peak-fit for sub-pixel accuracy strongly decreases for particle image diameter not equal 2-3 pixel. Before the optical aberrations can be reduced or eliminated, they have to be identified first. Using the PIV equipment this is easily possible as higher order aberrations become clearly visible by analysing the image of a regular grid, according to section 3.2, whereas the main primary aberrations can be easily observed by examination the image-symmetry of small particles within a thin light-sheet. For three different field-coordinates figure 4.9 shows the images of olive-oil droplets, with
è
¡
m, through a tilted BK7 glass-plate of constant thickness (10 mm). The most striking feature is the variation of the magnitude and direction of the dominant aberration by probing continually through focus and across the field of view.
When the optical system is perfectly aligned and the aberrations are below digital regis-tration, the diffraction limited image of a particle appears as a bright circular core surrounded by several rings of rapidly diminishing brightness. This is shown in the central column of figure 4.9 but the small variations of the intensity distribution are smeared out by the low res-olution of the used CCD sensor. As the object moves further off-axis, the diffraction limited particle image pattern alters from a bright central area surrounded by dark and bright rings to a dark central area surrounded by bright and dark rings as shown in the lower image. On the other hand, particles, located an appreciable distance apart from the optical axis deform into a line of certain orientation due to the tilted glass-window, see outer image pair of top row. By
1Distortion occurs when off-axis points are not formed at the location predicted by the paraxial equations.
4 Multiplane Stereo Particle Image Velocimetry
FIGURE 4.9: Images of tracer particles (¢ £¤ m) observed through a tilted glass-plate (BK7) of constant thickness 10 mm for three field locations as a function of the focus. The lines of constant intensity (isophote) are circular near the centre of the field but have a more complex form in the outer part of the image. The size of each sample is 128 pixelî .
changing the focus this line becomes elliptical and out-of-focus effects start to occur like the decreased brightness and the intensity gap in the centre of the particle-image, compare outer image pairs of figure 4.9.
The mentioned deformation is the main aberration the experimentalist has to deal with in praxis. It appears already when a non-collimated light-front enters or passes planar optical elements of different index of refraction, such as glass windows which separate the test-section from the laboratory, and it becomes even more pronounced for curved interfaces like lenses [6]. Although a detailed analysis of aberrations requires the theory of diffraction in order to account for the intensity distribution, the main features become evident by using the principles of geometrical optics which identify the image by the points of intersection of the geometrical rays with the image plane. Using this approximation, the starting point of this consideration is the law of refraction
ZCç]ê|ë2¥]¦çÙèsZmîgê|ë2¥§¦0î
which describes the direction of a light ray after diffraction at the planar interface between two homogeneous, isotropic media of differing index of refraction. To obtain the image position
n
of an object located on the optical axis at ¨ , this law has to be applied for each ray emerging from this point. Using the relations
¦ç è©«ª ç
and
¦0î è¬]ªCî
according to figure 4.10, the following formula can be derived.
n è
Zmî
¨
ZC篮êoª<ç h
±°
Zµç
Zî
ê|ë5¥§ª ç²
î
(4.7) This equation implies the appearance of aberration as the geometrical picture of an object-point does not possess a unique image object-point. For each aperture-angle
ª ç
there exists a certain
4.7 Monochromatic aberrations
FIGURE4.10: Imaging due to refrac-tion at a plane surface at finite aper-ture after [6].
intersection-width
n
whose exact location depends on the object-distance ¨ and the index of refraction of the two media. In addition, the magnitude of the aberrations is proportional to the angle of incidence, because the variation of the intersection width ³
n
increases with increasing aperture angle. The limit ¨µ´ ¶ is important as all rays intersect in only one point located at infinity and no aberrations occur at all. This situation can be generated either by using an optical collimator which forms parallel rays or by stopping the lens in a way that all rays are nearly parallel before they enter the planar interface. The last possibility is frequently used in PIV but its applicability mainly depends on the output energy of the laser, on the cross-section of the light-sheet, on the scattering behaviour of the particles and finally on the sensitivity of the CCD camera. It should be emphasised that in contrast to the refraction, reflections at a planar interface are aberration free, because all light rays from an object perfectly intersect in one virtual image point independent of the angle of incidence on the mirror. This is different for non planar surfaces where caustics can be observed.
Principle
FIGURE 4.11: Schematic representation of the image size and orientation for a non-axial object point as a function of the lateral position after [6].
When the situation is considered where an object point lies a considerable distance away from the optical axis, as indicated in figure 4.11, the incident cone of rays will strike the lens in an asymmetrical way. As a result, the focal length in this plane will be different as well due to the different optical path. In effect the meridional rays are tilted more with
4 Multiplane Stereo Particle Image Velocimetry
regard to the lens than the sagittal rays and they have a shorter focal length. This results in a significant change in symmetry as a function of the image distance. The initially circular cross section of the beam, emerging from the lens, becomes elliptically with the major axis in the sagittal plane and degenerates into a line (primary image) at the tangential or meridional focus.
Beyond this point the beam cross-section rapidly expands until it is again circular (circle of least confusion). Moving further away from the lens the beam cross-section again deforms into a line (secondary image) while the orientation of this line is rotated by½¿¾À relative to the primary image. This behaviour is similar to that shown in figure 4.9 and can be studied best by probing throughout the focus while observing non-axial particle images.
Aberrations introduced by means of the optical components within the laser, as the fre-quency doubler crystal or for the generation of the light-sheet, are more difficult to observe but of similar importance for PIV. Both systems have to be carefully aligned in order to obtain the best possible signal in the image-plane as information loss in this stage cannot be recon-structed, especially not by software solutions. The first testing procedure on any lens, after it
FIGURE 4.12: Top row: Intensity distribution of a Nd:YAG beam behind a tilted converging lens for three different angles of rotation. Bottom row: dependence of the light-sheet profile on the magnitude and direction of the aberration.
has been set up in the light-sheet-optics, is to rotate the optical elements about their own axis while examining the image. If there should be any de-centreing or tilt, lateral asymmetries in the point image will appear to rotate with the lens. This is shown in the top row of figure 4.12 for three different angles of rotation. The bottom row of the same figure reveals the same operation, but a fixed cylindrical lens was placed behind the rotating lens in order to high-light the direct dependence of the high-light-sheet profile from the magnitude and direction of the aberration. Generating an extremely thin light-sheet, which is necessary for high resolution
4.8 Feasibility study