- 12.1 INTRODUCTION. - Electronic cameras (vidicons) were first used as a recording medium in holography at the beginning of the 1970s. - At the beginning of the 1990s, computerized ‘reconstruction’ of the object wave was first demonstrated. - This is, however, not a reconstruction in the ordinary sense, but it has proven possible to calculate and display the reconstructed field in any plane by means of a computer. - The success of the CCD-camera/computer combi- nation has also prompted the development of speckle methods such as digital speckle photography (DSP).. - When used in DSP, the size σ s of the speckles imaged onto the target must be greater than twice the pixel pitch p, i.e.. - where α is the maximum angle between the object and reference waves and λ is the wavelength. - 12.2 TV HOLOGRAPHY (ESPI). - Therefore the reconstruction process is performed electronically and the object is imaged onto the TV target. - Because of the rather low resolution of a standard TV target, the angle between the object and reference waves has to be as small as possible. - This means that the reference wave is made in-line with the object wave. - When the system in Figure 12.1 is applied to vibration analysis the video store is not needed. - As in the analysis in Section 6.9, assume that the object and reference waves on the TV target are described by. - u o = U o e iφ o (12.4a). - Figure 12.1 TV-holography set-up. - u = U e iφ (12.4b). - U 2 + U o 2 + 2U U o cos[φ − 2kD(x) cos ωt] (12.6) This spatial intensity distribution is converted into a corresponding time-varying video signal. - When the vibration frequency is much higher than the frame frequency of the TV system ( 25 1 s, European standard), the intensity observed on the monitor is proportional to Equation (12.6) averaged over one vibration period, i.e. - I = U 2 + U o 2 + 2U U o cos φJ 0 (2kD(x)) (12.7) where J 0 is the zeroth-order Bessel function and the bars denote time average. - In the filtering process, the first two terms of Equation (12.7) are removed. - (12.8) Actually, φ represents the phase difference between the reference wave and the wave scattered from the object in its stationary state. - Equation (12.8) is quite analogous to Equation (6.51) except that we get a | J 0 | -dependence instead of a J 0 2 -dependence. - The maxima and zeros of the intensity distributions have, however, the same locations in the two cases. - A time-average recording of a vibrating turbine blade therefore looks like that shown in Figure 12.2(a) when applying ordinary holography, and that in Figure 12.2(b) when applying TV holography. - We see that the main difference in the two fringe patterns is the speckled appearance of the TV holography picture.. - When applied to static deformations, the video store in Figure 12.1 must be included.. - This could be a video tape or disc, or most commonly, a frame grabber (see Section 10.2) in which case the video signal is digitized by an analogue-to-digital converter. - Assume that the wave scattered from the object in its initial state at a point on the TV target is described by. - u 2 = U o e i(φ o +2kd) (12.10). - Figure 12.2 (a) Ordinary holographic and (b) TV-holographic recording of a vibrating turbine blade. - where d is the out of plane displacement and where we have assumed equal field ampli- tudes in the two cases. - I 1 = U 2 + U o 2 + 2U U o cos(φ − φ o ) (12.11) where U and φ are the amplitude and phase of the reference wave. - This distribution is con- verted into a corresponding video signal and stored in the memory. - I 2 = U 2 + U o 2 + 2U U o cos(φ − φ 0 − 2kd) (12.12) These two signals are then subtracted in real time and rectified, resulting in an intensity distribution on the monitor proportional to. - (12.13) The difference signal is also high-pass filtered, removing any unwanted background signal due to slow spatial variations in the reference wave. - Apart from the speckle pattern due to the random phase fluctuations φ − φ 0 between the object and reference fields, this gives the same fringe patters as when using ordinary holography to static deformations. - In the first place, the cumbersome, time-consuming development process of the hologram is omitted. - TV holography is extremely useful for applications of the reference wave modulation and stroboscopic holography techniques mentioned in Section 6.9. - When analysing static deformations, the real-time feature of TV holography makes it possible to compensate for rigid-body movements by tilting mirrors in the illumination beam path until a minimum number of fringes appear on the monitor.. - 12.3 DIGITAL HOLOGRAPHY. - In ESPI the object was imaged onto the target of the electronic camera and the interference fringes could be displayed on a monitor. - We will now see how the image of the object can be reconstructed digitally when the unfocused interference (between the object and reference waves) field is exposed to the camera target. - We assume the field amplitude u o (x, y) of the object to be existing in the xy-plane. - Let the hologram (the camera target) be in the ξ η-plane a distance d from the object. - Assume that a hologram given in the usual way as (cf. - u o | 2 + ru ∗ o + r ∗ u o (12.14) is recorded and stored by the electronic camera. - Here u o and r are the object and reference waves respectively. - Figure 12.3. - To find the reconstructed field amplitude distribution u a (x , y ) in the x y -plane we therefore apply the Rayleigh–Sommerfeld diffraction formula (Equation (4.7)):. - ρ cos dξdη (12.15). - ρ = d 2 + (ξ − x ) 2 + (η − y We therefore should be able to calculate u a (x , y ) in the x y -plane at any distance d from the hologram plane. - d where the virtual image is located (see Section 6.4), and (2) d = d, the location of the real image, provided the reference wave is a plane wave. - With today’s powerful computers it is straightforward to calculate the integral in Equation (12.15). - However, with some approximations and rearrangements of the inte- grand, the processing speed can be increased considerably. - The first method for solving Equation (12.15) is to apply the Fresnel approximation as described in Section 1.7. - 2iπ(νξ + µη)} dξ dη (12.17) where we have introduced the spatial frequencies. - dλ (12.18). - Equation (12.17) can be written as. - u a = z F { I · r · w } (12.19). - (12.20). - (12.21) In most applications z(u, v) can be neglected, e.g. - (12.22) By using this as the reconstruction wave, r · w = constant, and again we get a pure Fourier transform. - In Figure 12.4 the Fresnel method is applied.. - In the second method we first note that the diffraction integral, Equation (12.15), can be written as. - I (ξ, η)r(ξ, η)g(x , y , ξ, η)dξ dη (12.23) where. - g(x − ξ, y − η) and therefore Equation (12.23) can be written as a convolution. - g (12.25). - F { I · r } F { g } (12.26) By taking the inverse Fourier transform of this result, we get. - (12.27). - (12.28) and therefore. - G } (12.29) which saves us one Fourier transform.. - Figure 12.4 Numerical reconstruction of the real image using the Fresnel method. - The bright central spot is due to the spectrum of the plane reference wave. - The object was a 10.5 cm high, 6.0 cm wide white plaster bust of the composer J. - An important application of digital holography is in the field of holographic interferometry. - Standard methods (see Chapter 6) rely on the extrac- tion of the phase from interference fringes. - Digital holography has the advantage of providing direct access to phase data in the reconstructed wave field. - u a = Ue iϕ (12.30). - Re { u } (12.31). - By reconstructing the real wave of the object in states 1 and 2 (e.g.. - 12.4 DIGITAL SPECKLE PHOTOGRAPHY. - The intensities I 1 and I 2 in the first and second recording we wrote as I 1 (x, y. - This could be done because we assumed the speckle displacement to be uniform within the laser beam illuminated area and for simplicity we assumed the displacement to be in the x-direction. - e i2π ud (12.32) Now we discuss another technique called Digital Speckle Photography (DSP). - J 1 · J 2 | 1−α (12.33) By using the result from Equation (12.32), we get. - 2α e i2π ud (12.34) To this we apply another Fourier transform operation (step 3):. - 2α e − i2π[u(ξ−d)+vη] dudv = G α (ξ − d, η) (12.35) where. - 2α e −i2π(uξ+vη) dudv (12.36) In practice, G α (ξ − d, η) emerges as an expanded impulse or correlation peak located at (d, 0) in the second spectral domain. - The parameter α controls the width of the correlation peak. - Here a plane of interest in the material is seeded with grains of an X-ray absorbing material and a speckled shadow image is cast on the X-ray film.
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