All of the above results have been considered using the binarization method from equation 4.1. This gives a standard square wave pattern which in the Fourier plane consists of very structured regular higher order terms. When these terms are partially or wholly passed through the spatial filter they cause the presence of similarly well structured residual aberrations which greatly contaminate the null interferogram. Depending on the spatial filter size these can reduce the residual RMS of the null interferogram to worse than λ/50 RMS. Whilst this
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Figure 4.13: Surface plot of the size of the first 20 Zernike terms and how they grow with increasing amount of astigmatism being displayed on the hologram using a spatial filter of F=150.
is not an especially poor level of accuracy it is worth looking into whether or not this effect can be reduced so that spatial filter size considerations can be somewhat more relaxed.
An alternate method of programming the hologram is to use a random bi- narization algorithm, Φ(x, y) = π if cos(φ) ≥ Random−1 : 1, 0 else. (4.2) The effect on the hologram that this binarization algorithm has can be seen in figure 4.14. Completely absent is the presence of the neat diffraction grating seen in a standard binarization however this pattern still results in a first order diffracted term corresponding to the desired aberration. Where the standard
Figure 4.14: Two holograms of the same aberration binarized using the standard normal method given in equation 4.1 on left, and using a random binarization given in equation 4.2
binarization technique gives a series of well structured higher order diffracted terms however the random technique does not. Instead it introduces a degree of background noise into the PSF which is devoid of regular structure so that passing this through a filter will introduce a random error component rather than structured aberrations. As the dominant source of error in the null inter- ferograms simulated above was due to contamination by structured harmonics it is worth quantifying the accuracy of a technique of producing holograms which may minimise the presence of these artifacts.
Figure 4.15 shows how the growth of the spatial filter size affects the pro- duction of 5 waves of Zernike astigmatism. Included in this is an intermediary binarization technique denoted as quasi-random where for each pixel, during the phase comparison step of the binarization it is either compared against a value of 0 as in equation 4.1 or a random value as per equation 4.2. The determination of this is once again decided using a random number generation function. This
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Figure 4.15: The growth of 5 waves Zernike Astigmatism with increasing spatial filter size for three different binarization techniques.
quasi-random technique was investigated as it produces holograms that contain a higher degree of structure than a fully random technique and therefore lessens the degree of background noise however also disrupts the structure of higher order harmonics. Figure 4.15 shows that the growth of the astigmatic term is slightly larger for a standard binarization technique, however all three methods plateau at a similar spatial filter size.
As mentioned previously however it is more important to consider the effect of the residual errors in hologram production in the case of null interferograms. Figure 4.16 shows the growth of the residual RMS wavefront error with increas- ing spatial filter size in the case of a null interferogram, with the aberration being nulled again being 5 waves Zernike astigmatism similar to figure 4.9. For all values of spatial filter size the effect of higher order structured terms pro-
Figure 4.16: The growth in residual error with increasing spatial filter size of a null interferogram involving the production of 5 waves Zernike astigmatism for three binarization methods.
duces a smaller error than the random background noise introduced through randomization. We also see from this figure that there is a slight increase in the growth of the randomised residual RMS as a function of increasing spatial filter size compared to the standard technique. In the case of the latter, a larger spatial filter passes through a greater amount of the higher structured harmon- ics, however for the randomized technique this increase will be a result of a larger spatial filter allowing a greater amount of randomised background noise through the system. It can be inferred from this that the errors induced from introducing a degree of random background noise causes a larger error growth than the allowing of structured orders to partially or fully pass does.
Figure 4.17 shows the growth of the null interferogram residual with increas- ing aberration size for a fixed spatial filter of 300 pixels, which corresponds to the results seen in figure 4.12. In the latter we see that increasing aberration
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Figure 4.17: Growth of residual RMS error in null interferograms for the random binarization technique with increasingly large aberrations on the SLM with a fixed spatial filter size of 300 pixels.
size causes variations in the residual RMS measurement which is understood to be an effect of varying amounts of higher order harmonics being passed through the filter contaminating the image. In the case of the random binarization tech- nique we see that the there is a roughly constant level of residual error for all but the largest sized aberrations simulated, at which point an increase is observed for the production of trefoil at the point where the PSF is no longer being fully transmitted through the Fourier plane, something also seen in figure 4.12. The lack of increase in these residuals with increasing aberration is again ascribed to the lack of higher order contaminants being passed through the filter.
Looking at the nulled phase maps for the fully random binarization tech- nique compared to standard, figure 4.18, helps to confirm that the erroneous aberrations we see in the case of figure 4.10 are a result of the presence of higher
Figure 4.18: Residual phase maps recovered from a simulated nulled interfero- gram of 5 waves astigmatism with a spatial filter size of F=300 for Left) standard binarization and Right) random binarization. Whilst the highly structured el- ements are eliminated by using a randomised technique the background noise introduced fluctuates on a similar scale to the height of the well defined astig- matic artifacts.
order structure transmitted through the spatial filter as they do not appear here when this structure is disrupted. It would appear from these results that using a non-random binarization technique to encode the hologram is preferable how- ever and should be used experimentally. Initial results found experimentally should be obtained without serious consideration to the spatial filter size, as residual errors even for the largest spatial filter sizes simulated are still in the area of 0.05 waves RMS and provisionally acceptable. After an experimental indication of SLM accuracy is obtained the effect of spatial filter size can be considered when attempting to reduce errors in the experiment.
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