- In the years following the advent of the laser, this pattern was considered a mere nuisance, especially in holography (and it still is. - In this chapter the basic principles of the different techniques of speckle metrology will be described. - This white-light speckle photography is included in the final section of the chapter.. - In Figure 8.1, light is incident on, and scattered from, a rough surface of height variations greater than the wavelength λ of the light. - Figure 8.1 Light scattering from a rough surface. - Figure 8.2 Photograph of a speckle pattern. - Figure 8.3 shows a plot of P I (I. - SPECKLE SIZE 195 A measure of the contrast in a speckle pattern is the ratio C = σ I / I , where σ 1 is the standard deviation of the intensity given by. - From Figure 8.2 we see that the size of the bright and dark spots varies. - To find a representative value of the speckle size, consider Figure 8.4, where a rough surface is illuminated by laser light over an area of cross-section D. - For simplicity, we consider only the y-dependence of the intensity. - Let us assume that the speckle pattern at P is a superposition of the fringe patterns formed by light scattered from all point pairs on the surface. - The period of this pattern is a measure of the smallest objective speckle size σ o which therefore is. - For smaller separations l, there will be a large number of point pairs giving rise to fringes of the corresponding frequency. - Figure 8.4 Objective speckle formation. - The calculation of the size of the resulting so-called subjective speckles is analogous to the calculation of the objective speckle size. - Here the cross-section of the illuminated area has to be exchanged by the diameter of the imaging lens. - Figure 8.6 Subjective speckle formation. - where m = (b − f )/f is the magnification of the imaging system. - However, even an ideal lens will not image a point into a point but merely form a intensity distribution (the Airy disc, see Section 4.6) around the geometrical image point due to diffraction of the lens aperture. - After development this negative is placed in the object plane of a set-up for optical filtering like that in Figure 4.12. - Figure 8.7 shows (Fourney 1978). - (2) the resulting diffraction pattern (the spatial frequency spectrum) in the x f , y f -plane. - and (3) typical form of the smoothed intensity distribution along the x f -axis.. - The dark spot in the middle of Figure 8.7(b) is due to the blocking of the strong zeroth- order component. - The same spectrum (as in Figure 8.8(b)) would have resulted by filtering out the first side. - Figure 8.7 (a) Speckle pattern and its corresponding. - Figure 8.8 Spectrum of a speckle pattern (a) before and (b) after filtering. - where f is the focal length of the transforming lens and λ 0 is the wavelength of the light source applied in the filtering process. - What we have actually done therefore is to select one of the numerous gratings with a continuum of directions and frequencies that constitute the laser-illuminated object.. - When the object undergoes an in-plane deformation, the speckle pattern will follow the displacements of the points on the object surface.. - By means of optical filtering of the double-exposed negative we get an intensity distribution dependent on the modulation function only (see Section 4.7, Equation (4.66. - The speckle pattern represents gratings of all orientations in the plane of the object (cf.. - Highest sensitivity is obtained by placing the filtering hole at the edge of the spectrum. - With a magnification m = 1, this gives a sensitivity equal to the laser wavelength mul- tiplied by the aperture number of the imaging lens. - Figure 8.9 shows an example of such Fourier fringes obtained by this method (Hung 1978). - Figure 8.9 Fringe patterns depicting the horizontal and vertical displacements of a cantilever beam obtained from the various filtering positions in the Fourier filtering plane. - A double-exposed negative of two speckle patterns resulting from a deformation therefore will consist of identical point pairs separated by a distance equal to the deformation times the magnification of the imaging system.. - When the beam covers one pair of identical points, they will act in the same way as the two holes P 1 and P 2 in the screen of the Young’s interferometer (see Figure 3.13, Section 3.6.1). - The situation is sketched in Figure 8.10. - If the displacement on the object is equal to s the separation of the corresponding speckle points on the negative is equal to m · s where m is the magnification of the camera. - By measuring the fringe separation d we can therefore find the object displacement at the point of the laser beam incidence using Equation (8.18). - Figure 8.10 Young fringe formation. - To obtain such a Young fringe pattern, the identical pairs of speckles must be separated by a distance which is at least equal to one half of the speckle size, that is 1/2σ s . - Figure 8.11 Young fringes at different points in a plate under tension in a miniature rig. - Figure 8.11 shows an example of the results obtained by this method. - This displacement can be represented by the peak position of the cross-correlation function c I X between the intensity distributions I 1 (x, y) and I 2 (x, y) (specklegram 1 and 2) of the speckle patterns before and after the object displacement.. - Physically, the correlation process can be visualized as the sliding of specklegram 1 over specklegram 2 and an assessment of the similarity between I 1 and I 2 for each value of the lag. - averaging over the whole area, and repeating for different values of the lag:. - For comparison, the autocorrelation (I 1 = I 2 ) of the pattern before translation is shown in Figure 8.12(a). - The peak of the autocorrelation is always located at zero. - The peak of the cross-correlation corresponds to the speckle displacement and the decrease in peak height is associated with change in the structure, so-called decorrelation.. - To analyse the laser speckle phenomenon further, we have to specify the statistics of the amplitudes of the speckle field. - u(x 1 , y 1 )u ∗ (x 2 , y is the autocorrelation function of the fields, also referred to as the mutual intensity.. - P (ξ 1 )P ∗ (ξ 2 )δ(ξ 1 − ξ where P (ξ ) is the amplitude of the incident field. - Thus the mutual intensity of the observed field depends only on the difference of the coordinates in the xy-plane. - Finally, the auto- correlation function of the speckle intensity assumes the form. - Another quantity of considerable interest is the power spectral density W (u, v) of the speckle intensity distribution. - When the ξ η-plane is imaged onto the xy-plane by a lens, and provided the object illu- mination is uniform, the size of the speckles incident on the lens pupil is very small compared to the size of the lens pupil. - Then, to a good approximation, the mutual inten- sity of the field in the lens pupil is given by Equation (8.26) and P (ξ, η) is the pupil function of the lens. - From the definition of the optical transfer function H (u, v) (Section 4.6, Equation (4.48. - in the observation plane is given by the Fourier transform of the transmittance, i.e.. - 2 forms an envelope of the Young fringes described by (1 + cos 2π ud).. - I a ∝ H (u, v)(1 + cos 2π ud) (8.42) We see that the Young fringes are modulated by the lens’s MTF: see Figure 8.13.. - Figure 8.13 Young’s fringes modulated by the lens MTF. - In Section 12.4 (Digital speckle photography) we shall see that the speckle displace- ment is measured by detecting the position of the cross-correlation peak directly. - As we saw in Figure 8.12, lack of correlation decreases the height of the correlation peak. - (8.43) Here u 1 and u 2 are the field amplitudes of the light from the two interferometer arms, and the brackets mean averaging over time. - 2 is the spectral distribution of the light and that. - The principle of speckle-shearing interferometry, also termed shearography, is to bring the rays scattered from one point of the object into interference with those from a neigh- bouring point. - This can be obtained in a speckle-shearing interferometric camera as depicted in Figure 8.14. - The set-up is the same as that used in ordinary speckle pho- tography, except that one half of the camera lens is covered by a thin glass wedge.. - In that way, the two images focused by each half of the lens are laterally sheared with respect to each other. - Figure 8.14 Speckle-shearing interferometric camera. - By double exposure of the object before and after deformation, a speckle fringe pattern depicting φ of Equation (8.53) will be generated. - A convenient means for obtaining the shear is to place a Michelson interferometer set-up with a tilt of one of the mirrors in front of the image sensor. - By an ingenious set-up constructed by the group at Lule˚a University of Technology simultaneous observation of the out-of- plane displacement and slope in real time is made possible. - The set up is shown in Figure 8.15 where an electronic holography configuration is combined with a shearog- raphy configuration so that the displacement and slope interferograms are displayed in each half of the monitor. - Figure 8.17 shows another advantage of shearography. - A crack in the centre of the object becomes visible in the upper image due to the fact that shearography is highly sensitive to local variations in the deformation field. - In the holographic phase map of Figure 8.17(b), however, it is difficult or impossible to see the same crack.. - Finally we mention that speckle-shearing interferometry can be obtained by intentional misfocusing of the object. - Figure 8.15 Schematic for simultaneous measurement of out-of-plane displacement and slope.. - Reproduced by courtesy of The Optical Society of America and of H. - These spheres represent a random distribution of points containing all spatial frequencies up to the reciprocal of the smallest sphere separation. - This highest spatial frequency is therefore limited by the size of the glass spheres. - However, if the spheres are small enough, or photographed from a sufficient distance, the highest spatial frequency will be limited by the impulse response of the imaging lens, i.e. - the diameter of the Airy disc (see Section 4.6) in the same way as in laser speckle photography. - The main difference is that the detailed structure of the white light speckles is fixed on the surface, not in space as with laser speckles. - The depth of focus, however, varies inversely as the square of the lens aperture (see Section 4.6).. - These conflicting requirements are partly overcome by an ingenious modification of the camera made by Burch and Forno (1975). - Figure 8.18 shows a white light speckle camera assembly. - Figure 8.19 shows an example of the application of this camera. - Figure 8.20 shows the results of such an experiment. - A film plate was clamped to one side of the Plexiglas plate which was rubbed to get speckles. - Figure 8.18 White light speckle camera. - Figure 8.20 (a) Notched Plexiglas plate under tension in miniature rig. - Figure 8.20 (continued). - After development the film is placed in the optical filtering set-up (using an He–Ne laser) shown in Figure 4.14 a distance z from the filter plane.. - 8.3 When imaging a speckle pattern a mask consisting of a double slit as sketched in Figure P4.4 is placed in the plane of the exit pupil.
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