Alfredo Dubra's PhD thesis - Imperial College London

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2. Lateral shearing interferometer design f AC max f DC max f c f sampling Figure 2.2: Lateral shearing interferogram spectra produced by a glass wedge. The red circle in the center contains the DC term, while the other two circles contain the AC terms that carry the phase information. The indicated spatial frequencies are: the sampling frequency f sampling , the carrier frequency f c and the maximum DC and AC term frequencies fmax DC and fmax AC respectively. this is not an important limitation because we have an extra degree of freedom in the shear, that can be tuned later by sliding the wedge position along the optical axis of the system, as will be explained in section 2.7.1. Two conditions must be met to avoid aliasing when spatially sampling the interferogram. Firstly, the spectra of the AC and DC terms of the interferogram intensity must not overlap, as it is the case in the example shown in figure 2.2. This is ensured by choosing a carrier frequency f c (i.e. the wedges angle) larger than the sum of the highest spatial frequencies of the DC and AC terms, that is f DC max + f AC max < f c . (2.13) Secondly, according to the sampling theorem, the spatial sampling frequency must be equal to or larger than twice highest spatial frequency of the interferogram, that is f c + f AC max < f sampling /2. (2.14) If information on the spatial content of the tear topography were available, one could put figures into these conditions and estimate an adequate angle for the wedges and the 29
2. Lateral shearing interferometer design number of camera pixels. However, we do not have an established model to describe the dynamics of tear topography. Therefore, making no assumptions on the spatial content of the tear topography, the angles of the glass wedges for the interferometer were chosen so that the carrier frequencies of the interferograms were sampled at around 8 camera pixels per fringe, with around 450 samples across each interferogram. These values correspond to a tilt between the reflected wavefronts on the front and back surfaces of the wedge of 12 mrad, and are a compromise between having high spatial frequency modulation in the interferograms and reasonable sampling. We now consider the shear introduced by the wedges, for which we recall equation 2.3 I(⃗r) = I 0 (⃗r) + I 0 (⃗r + ⃗s) + 2 √ [ ] φ(⃗r + ⃗s) − φ(⃗r) I 0 (⃗r) I 0 (⃗r + ⃗s) cos s + 2πf s ⃗ c · ⃗r , where for convenience the phase difference has been multiplied and divided by the shear amplitude s. We can now see that if the shear is small in comparison with the spatial changes in the phase, the phase difference can be approximated by the phase slope in the shear direction multiplied by the shear amplitude. We can further assume that in terms of measured phase, the signal-to-noise ratio (SNR) is proportional to the shear amplitude. On the other hand, the larger the shear the smaller the area over which the pupils overlap, and so the poorer the phase estimation. Again, as with the tilt, an optimum estimation of the shear needed for the experiment would need some knowledge of the wavefronts to be measured, and therefore the choice of shear is arbitrary, and in our case is approximately 5 % of the beam diameter for a beam diameter of 3 mm. Finally, we will consider 2 % as an acceptable tolerance to the shear variation across the beam. Taking all the above in to consideration for a beam diameter of 3 mm, we reach the following values for the wedge angle δ and thickness: δ ≈ 10 arcmins, (2.15) thickness ≈ 0.4 mm, (2.16) The two glass wedges manufactured have angles 11 and 13 arcmins, and mean thicknesses 0.35 and 0.70 mm respectively. The extra thickness of the second wedge will only affect the shear amplitude, but as will be shown later (see section 2.7.1), this can be compensated for by displacing the wedge along the optical axis. Both glass wedges were produced to be used at angle of incidence α 1 of 45 o and made of BK7, 30
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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5. Preliminary experiments bb2 (t =
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5. Preliminary experiments dm1_no_w
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5. Preliminary experiments bb_focus
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5. Preliminary experiments the corr
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5. Preliminary experiments 1 10 1 2
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6. Tear topography dynamics experim
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Chapter 7 Summary and discussion Th
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7. Summary and discussion The value
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7. Summary and discussion ao2_radia
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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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