Alfredo Dubra's PhD thesis - Imperial College London

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4. Wavefront reconstruction from shear phase maps each of width 32 samples and with unit shear. There are two points to note from these plots. Firstly, the noise coefficient actually decreases with small data extension. This suggests that all the existing DFT methods could be improved by applying data extension. Secondly, we can see that extending the pupil around all sides leads to a smaller noise coefficient than keeping the pupil in a corner of the grid. This can be understood by considering the noise variance of the various difference data points. Recall that in order to use the DFT reconstruction methods, the gradient data need to be made to meet the periodicity condition, using equations 4.5 and 4.6. That is, for row (or column) of data in a subgrid, an extra data point given by the negative sum of the data in the row is added to ensure that the gradients along the row sum to zero (corresponding to a periodic phase function). The noise in the extra data is given by the sum of the noise in the row of data. If we consider Gaussian noise of variance σ 2 in the original data, then these extra data points have noise variance N x /s x σ 2 (N x /s x being the number of data points in the row of the subgrid). This suggests that for large grids, there will be a considerable noise contribution around the edge due to the extra periodization data. For the 2D reconstructor this will have greatest impact on the reconstructed phase near the edges. Extending the data pushes these edges further from the original pupil, reducing its effect in the reconstructed phase. This is illustrated in figures 4.6 (a) and (b), which plot the spatial distribution of the noise in the reconstructed phase maps (i.e. the noise variance) for square 32 × 32 sample pupils with no data extension and with 4 rows and columns (two around each side) data extension respectively. The plots correspond to the first and third circles of plot 4.5 (a). The plots of spatial distribution of noise variance in figures 4.6 a) and b) correspond to square pupils 32 samples across, unit shear, without data extension (first circle in figure 4.5 a) and with 2 columns/rows at the pupil sides (third circle in figure 4.5 a). 4.5 Conclusions We have presented two alternative methods to extend DFT-based wavefront reconstruction in a convex pupil from difference maps to cope with arbitrary problem dimensions and arbitrary integer shears, with performance comparable to that of the pseudoinverse in terms of the noise propagation coefficient. Non-optimized algorithm implementations showed significant speed increase with respect to the least squares 75
4. Wavefront reconstruction from shear phase maps N ext/2 N ext/2 Next N ext/2 N ext/2 Next 1.5 1.5 1.4 1.4 1.3 1.3 Noise coefficient 1.2 1.1 Noise coefficient 1.2 1.1 1 1 0.9 0.9 0.8 0 5 10 15 N ext 0.8 0 5 10 15 N ext (a) (b) Figure 4.5: Effect on noise coefficient of extending the data by enlarging a square (a) and a circular (b) pupil of 32 samples across and unit shear against N ext , the number of columns/rows of padding for extension from two sides of the grid only (dots) and for symmetric extension from all the four sides of the grid (circles). The noise coefficient of the pseudoinverse is shown as a solid line for reference. 6 6 5 5 4 4 3 3 2 2 1 1 0 30 0 25 20 15 10 5 5 10 15 20 25 30 30 25 20 15 10 5 5 10 15 20 25 30 (a) (b) Figure 4.6: Spatial distribution of noise variance for DFT methods for a cornered (a) and centered (b) square pupil of 32 samples across. 76
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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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1. Introduction A less significant
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1. Introduction It should also be m
Page 25 and 26: Chapter 2 Lateral shearing interfer
Page 27 and 28: 2. Lateral shearing interferometer
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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
Page 81 and 82: 5. Preliminary experiments bb2 (t =
Page 83 and 84: 5. Preliminary experiments dm1_no_w
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Page 87 and 88: 5. Preliminary experiments the corr
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Page 91 and 92: 6. Tear topography dynamics experim
Page 107 and 108: Chapter 7 Summary and discussion Th
Page 109 and 110: 7. Summary and discussion The value
Page 111 and 112: 7. Summary and discussion ao2_radia
Page 113 and 114: A. Preliminary study of feasibility
Page 125 and 126: 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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