Alfredo Dubra's PhD thesis - Imperial College London

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3. Data processing 3.1.3 Implementation details In practice the algorithm calculates the square modulus of the discrete Fourier transform (DFT) of the raw interferograms, and then performs the search of the maximum value over a semi-annular region centered at the zero spatial frequency, with the inner and outer radii corresponding to the limits of the range we expect the carrier frequency to have. This search range was set to spatial frequencies with wavelengths between 5 and 17 image pixels. The maximum of the AC term is only found within pixel accuracy, sub-pixel accuracy by spectra interpolation was not pursued. The only consequence of this coarse estimation is to leave a residual tip/tilt on the retrieved phase maps that is removed later. 3.1.4 Tests To illustrate the performance and limitations of the algorithms described in this chapter, we will show the results of their application to 8 interferograms that display the different features we have found in the almost 5000 usable interferograms. These results are by no means an exhaustive test, although they provide a good sample of all the possible scenarios that could be found. Figure 3.1 shows a series of raw interferograms and the corresponding frontal view of the 3 dimensional plot of the calculated spectra. The region of the spectra within the red vertical lines was excluded from the search for the maxima (inner diameter of the maximum search area), and the DC term can be clearly recognized in it for all figures. However, the AC terms in the far left and right of the spectra plots can be difficult to identify in some of the interferograms, because when the phase or intensity of the interferogram contains rapid variations, the spectra of the DC and AC term broaden and overlap, as can be seen in figures 3.2 b), f) and h). These figures illustrate two very important problems that are critical for automated processing of the data. First, the algorithm will always find a maximum which might not be related to the modulation frequency, but to other features on the interferograms, such as the speckle grain size in figures 3.2 f) and h). Second and more important is the issue of the spectra overlapping, since the method we use to retrieve the phase of the interferogram is based on the assumption that the DC and AC spectral terms do not overlap. When this is not the case there will be a non-linear influence of the DC 43
3. Data processing term on the retrieved phase. We should therefore either visually inspect the spectra of all the interferograms to be processed, or find an automated method to decide on whether or not we consider the DC and AC spectra do overlap, to discard them as not usable 1 . 3.2 Shear estimation 3.2.1 Theory Let us assume that we have an image that consists of the sum of the intensities of two copies of the same object A (represented by a triangle in figure 3.3) shifted with respect to each other by a vector ⃗s. The normalised autocorrelation C AA of this image can be defined as ∫ ∫ [ A( r ⃗′ ) + A( ⃗ [ r ′ + ⃗s)] A( r ⃗′ − ⃗r) + A( ⃗ ] r ′ + ⃗s − ⃗r) d 2 r ′ C AA (⃗r; ⃗s) = ∫ ∫ [ A( r ⃗′ ) + A( r ⃗ ] 2 (3.1) ′ + ⃗s) d 2 r ′ where we assume that A is a real function, and in practice will only take non-zero values over a finite range. For an autocorrelation, we expect to get a maximum value when ⃗r is zero, but in this particular case, we will also find that C AA has two other peaks at ⃗r = −⃗s and ⃗r = ⃗s. These peaks correspond to the situation illustrated in figure 3.3 b) and c) respectively. Thus, the position of the side peaks are at ± the shear. This is the idea behind the algorithm used to estimate the shear between the pupils in the shearing interferograms. 3.2.2 Algorithm The shearing interferograms are more complex than the illustration of figure 3.3, they also contain the interference (AC) term. The autocorrelation of such interferograms, has multiple side fringes with the same spacing as the fringes in the interferogram as illustrated by figure 3.4 (a) and (b). If we low-pass filter the raw interferogram (c) to remove the AC term, one could resolve central peak and two side peaks (f). Although 1 If we were to automate the evaluation of the spectra overlapping, we could use as a first step the second order moment of the area around the estimated maximum, as defined in section 6.5 and then setting a threshold value to determine when the AC term corresponds to a relatively smooth topography. 44
Page 1 and 2: A Shearing Interferometer for the E
Page 3 and 4: Acknowledgements Pursuing the PhD d
Page 5 and 6: Contents Contents 4 List of Figures
Page 7 and 8: CONTENTS 5.1 Prydal type experiment
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Page 13 and 14: Chapter 1 Introduction Assessing an
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7. Summary and discussion The value
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A. Preliminary study of feasibility
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B. Laser safety calculations B.1 MP
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Bibliography [1] T. Young. The Bake
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BIBLIOGRAPHY [22] P. Artal, S. Marc
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BIBLIOGRAPHY [44] Austin Roorda and
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BIBLIOGRAPHY [67] W.N. Charman and
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BIBLIOGRAPHY [89] J.P. Craig, P.A.
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BIBLIOGRAPHY [115] Frangois C. Delo
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Alfredo Dubra's PhD thesis - Imperial College London

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