Alfredo Dubra's PhD thesis - Imperial College London

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4. Wavefront reconstruction from shear phase maps by the number of copies required for the extension. This suggests that unnecessary data extension should be avoided. 4.2.4 Filter poles and subdivision method In this section, addressing of filter poles problem in equation 4.4 will lead us to an alternative method for approaching the reconstruction problem when the dimensions of the original grid do not meet the divisibility condition. From equations 4.1 and 4.2, it can be seen that there are some wavefronts, with period s x × s y , which give rise to zero for all the differences. These define the null space and correspond to the poles of the reconstruction filter (equation 4.4). By setting the values of the filter at the poles (which occur when ks x /N x and ls y /N y are integers) to zero, one eliminates only these modes unsensed by the equations defining the forward problem, thus, producing a solution with minimum norm. Therefore, we replace the reconstruction filter of equation 4.4 by ⎧ ⎪⎨ 0 when k sx [ ( ) [ ( ) N x ˜φ(k, l) = exp −2πi −1] ⎪⎩ k sx ˜Dx (k,l)+ exp −2πi −1] l sy ˜Dy (k,l) Nx [ ) )] Ny elsewhere. 4 sin 2( π k sx Nx +sin 2( π l sy Ny and l sy N y are integers (4.8) Equivalently, we can see that sample points in the original problem are only linked by difference data if they are separated by multiples of the shear distances s x , s y . The original sample grid can by split into s x × s y interlaced subgrids with sample spacing s x , s y which can be treated independently from each other, since points on different subgrids are not linked by difference data (figure 4.2). Thus, the original reconstruction problem can be subdivided into s x × s y independent problems each of which can be treated as having shears s x = s y = 1, and which can therefore be solved by direct application of equation 4.8. The resulting phase maps are re-interlaced onto the original grid. Since there is no information about the relative piston term of these interlaced grids in the original measurements, the piston term (i.e. the mean phase over the pupil) for each of the reconstructed subgrids is set to zero. It is equivalent to setting the null mode terms to zero, resulting in the minimum norm solution. We shall refer to this method as subdivision. 71
4. Wavefront reconstruction from shear phase maps x y subgrid (1,1) subgrid (2,1) subgrid (3,1) subgrid (1,2) subgrid (2,2) subgrid (3,2) Figure 4.2: Example of the interlaced subgrids for a grid of dimensions N x = 6 by N y = 7 and shears s x = 3 and s y = 2. The relative phases between points in different subgrids is undetermined by the data. 4.3 Computing performance An important consideration when comparing alternative reconstruction methods is their computational requirements. Figure 4.3 a) plots the number of real multiplications required to perform the wavefront reconstruction over a square grid with a total of N points and shear s
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A Shearing Interferometer for the E
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Acknowledgements Pursuing the PhD d
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Contents Contents 4 List of Figures
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CONTENTS 5.1 Prydal type experiment
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List of Figures 1.1 Young’s exper
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LIST OF FIGURES 5.2 Sequence of ref
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Chapter 1 Introduction Assessing an
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1. Introduction (a) (b) Figure 1.3:
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1. Introduction was then modified a
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1. Introduction been achieved by us
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Page 25 and 26: Chapter 2 Lateral shearing interfer
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Page 67 and 68: Chapter 4 Wavefront reconstruction
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Page 79 and 80: Chapter 5 Preliminary experiments 5
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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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