- Figure 3.2 is an illustration of two interfering plane waves. - This is an example of the moir´e effect and the resulting fringes are called moir´e fringes or a moir´e pattern. - Figures 3.4, 3.8 and 3.9 are examples of the same effect. - Today there is little left of the moir´e effect, but techniques applying gratings and other type of fringes are widely used. - (7.1) where p is the grating period and where 0 <. - This means that the grating given by Equation (7.1) can be expressed as t 2 (x, y. - (7.2) ψ (x) is the modulation function and is equal to the displacement of the grating lines from its original position divided by the grating period. - When the two gratings given by Equations (7.1) and (7.2) are laid in contact, the resulting transmittance t becomes the product of the individual transmittances, viz.. - 1 + cos π ψ (x) cos 2π x. - p + 1 2 ψ (x) (7.5) Here we see that the term cos π ψ (x) describing the moir´e fringes are amplitude modu- lating the original grating.. - Equation (3.21)). - I 0 + I 1 cos π ψ (x) cos 2π x. - When applying low-frequency gratings, all these methods may be sufficient for direct observation of the modulation function, i.e. - When using high-frequency gratings, however, direct observation might be impossible due to the low contrast of the moir´e fringes. - The angle of incidence and grating period are adjusted so that the direction of the first diffracted side order coincides with the object surface normal. - the measuring sensitivity) is limited by the resolving power of the imaging lens. - Surface curvature might also be a problem when using methods (3), (4) and (5) because of the limited depth of focus of the imaging lens. - By using one of these methods, we will, either directly or by means of optical filter- ing, obtain an intensity distribution of the same form as given in the two first terms in Equation (7.11) or (7.12). - Figure 7.2(a) shows an example of such an intensity distribution with the corresponding displacement and strain in Figures 7.2(b) and (c).. - A lot of effort has therefore been put into increasing the sensitivity of the different moir´e techniques (G˚asvik and Fourney 1986). - Figure 7.2 (a) Example of the intensity distribution of a moir´e pattern with the corresponding;. - An analysis of such gratings would have resulted in expressions for the intensity distri- bution equivalent to Equations (7.11) and (7.12), but with an infinite number of terms containing frequencies which are integral multiples of the basic frequency. - When using such gratings it is therefore possible to filter out one of the higher-order terms by means of optical filtering. - By filtering out the Nth order, one obtains N times as many fringes and therefore an N-fold increase of the sensitivity compared to the standard technique.. - However, the intensity distribution of the harmonic terms generally decreases with increasing orders which therefore sets an upper bound to the multiplication process. - CONTOURING 179 a photograph of the fringes. - When forming the reference grating by interference between two plane waves, interpo- lation can be achieved by moving the phase of one of the plane waves. - a quarterwave plate and a rotatable polarizer in the beam of the plane wave.. - Instead of counting fringe orders due to the deformation, one measures the deviation or curvature of the initial pattern. - The principle of the method is shown in Figure 7.3.. - This is equivalent to a displacement of the grating relative to its shadow equal to. - u = u 1 + u 2 = h(x, y)(tan θ 1 + tan θ where h(x, y) is the height difference between the grating and the point P 1 on the surface. - Figure 7.3 Shadow moir´e. - This is a good solution, especially for large surface areas which are impossible to cover with a plane wave because of the limited aperture of the collimating lens.. - If the surface height variations are large compared to the grating period, diffraction effects will occur, prohibiting a mere shadow of the grating to be cast on the sur- face. - Perhaps the most successful application of the shadow moir´e method is in the area of medicine, such as the detection of scoliosis, a spinal disease which can be diagnozed by means of the asymmetry of the moir´e fringes on the back of the body. - Takasaki has worked extensively with shadow moir´e for the measurement of the human body. - Figure 7.4 shows an example of contouring of a mannequin of real size using shadow moir´e.. - Figure 7.5 shows fringes with an inter-fringe distance d projected onto the xy-plane under an angle θ 1 to the z-axis. - Figure 7.4 Shadow moir´e contouring. - Figure 7.5 Fringe projection geometry. - From Equation (7.3) this gives a modulation function equal to. - cos θ 2 (7.21). - This can be achieved by means of a Twyman-Green interferometer with a small tilt of one of the mirrors (Figure 7.6(a)).. - 2 sin(α/2) (7.22). - Figure 7.6 Fringe projection by means of (a) interference and (b) grating imaging. - Figure 7.7 Grating projection. - Figure 7.6(b) shows another method for projecting a fringe pattern onto the surface.. - This situation can be analysed more closely from Figure 7.7 where a light ray through the centre of the projection lens goes from point A on the grating to point B on the xy-plane. - A lies a distance s from the optical axis of the projection lens and B is a distance x from the origin of the coordinate system. - (7.25) where l p is the distance from the lens to the origin of the coordinate system. - where a is the grating–lens distance, we get x = x(s. - (7.27) Equation (7.27) gives the position x = x(s) as a function of the position s on the grating.. - l p + x sin θ o (7.30). - We find the phase ϕ in the xy-plane from Equation (7.30):. - where f o is given by Equation (7.36). - A + B cos 2π ϕ (7.38). - with ϕ given from Equation (7.37). - From the definition of the instantaneous frequency, we get. - Figure 7.8 Fringe projection geometry. - When the camera is pointing along the z-axis, we see from Figure 7.8 that tan θ 1 = l p sin θ o + x. - z cos θ o sin θ o + (l k − l p cos θ o )x l p l k (7.42) ψ (x. - the contour interval becomes independent of the position on the surface) if the projection lens and the camera lens are placed at equal heights above the xy-plane (l k − l p cos θ o = 0).. - Assume that a point on the surface in Figure 7.5 executes harmonic out-of-plane vibra- tions given by. - z = z 0 + a cos ωt (7.46). - The intensity distribution of the projected pattern (cf. - The expression can be written as. - φ t = (2π/d) sin θ a cos ωt (7.49b). - By photographing this pattern with an exposure time much longer than the vibration period T , the resulting transmittance t of the film becomes proportional to I (x, t) averaged over the vibration period. - 2π d a sin θ (7.51) which inserted into Equation (7.50) gives. - 2π d a sin θ e iφ c (7.53). - 2π d a sin θ (7.54) From the values of the arguments of the Bessel function corresponding to its maximum and zeros given on page 167, we find that light fringes occur when. - sin θ (7.56). - which is a figure representing the sensitivity of the method.. - A set-up for projection moir´e is shown in Figure 7.10 where a grating is projected onto the object. - Figure 7.11 shows some examples of the results obtained with such a system.. - Figure 7.11(a) shows a cartridge casing with a dent. - The reference image stored into the memory is taken from the undefective side of the casing. - Figure 7.11(b) shows the result from two recordings of a 25-litre oil can before and after filling with water. - In Figure 7.11(c) the system is applied to vibration analysis. - The picture is a time-average recording resulting in a zeroth-order Bessel fringe function displaying the amplitude distribution of the plate as described in Section 7.5.3. - Figure 7.10 Projection moir´e using TV-camera and digital image processor. - Figure 7.11 Examples of TV-moir´e fringes: (a) cartridge casing with a dent. - Most of them are, however, variations of the basic principles discussed in the preceding sections. - Figure 7.12 Reflection moir´e. - Figure 7.12 shows the principle of the method. - The smoothness of the surface S makes it possible to image the mirror image of the grating G by means of the lens L. - The result is a moir´e pattern defining the derivative of the height profile, i.e. - the slope of the deformation.. - In an analysis of the resolution of the reflection moir´e method it is found that the maximum resolution that can be obtained with a viewing camera is of the order 7 × 10 −3 radians.. - In Figure 7.13 a laser beam is incident on a diffusely scattering surface under an angle θ 1 . - The optical axis of the lens makes an angle θ 2 to the surface normal. - Figure 7.13 Triangulation probe. - the corresponding movement of the imaged spot on the detector is given by (see Eq. - cos θ 1 = ms(tan θ 1 cos θ 2 + sin θ where m is the transversal magnification of the lens. - it gives an output voltage proportional to the distance of the light spot from the centre of the detector. - It is the centroid of the light spot that is sensed and thus the position measurement is independent of the spot diameter as long as it is inside the detector area. - However, the size of the light spot will also be magnified, and this must always lie inside the detector area to avoid measurement errors, thus limiting the usable magnification.. - (a) Write down the transmittances t 1 and t 2 of the grating before and after the load.. - (c) If ε x = 10 −3 , the grating frequency is 20 lines/mm and the length of the bar is 10 cm, how many moir´e fringes are observed?
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