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February 26, 2018

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A traditional error budget aims at quantifying the deterioration of the contrast with the root mean square(rms) error phase applied on the segments. For example, in the case of segment-level pistons, we can easily deduce from Fig.1 the constraints in piston co-phasing in term of rms error. For example, in the case of segment-level pistons, we can easily deduce from Fig.1 the constraints in piston co-phasing in term of rms error.

Contrast as a function of the rms piston error phase
Fig.1. Contrast as a function of the rms piston error phase on the pupil, computed from both the end-to-end simulation (E2E) and PASTIS.


Since PASTIS provides an accurate (~3% error) estimation of the contrast, but 10^7 times faster than the end-to-end simulation, it can replace this very time-consuming method in such error budgeting, which is particularly useful when numerous cases need to be tested. Similarly, it makes simulations of performance for long-time series of high-frequency vibrations possible.

However, it is known that some segments have a bigger impact on the contrast than others, which appears in the PASTIS model. This is why we propose another approach to error budgeting, which provides also a better understanding of the repartition of the requirements on the segments. 

First, from PASTIS we can derive the eigen modes of the pupil. Some of them are shown in Fig. 2, in the piston case. Since these eigen modes are orthonormal, they provide a modal basis of the segment-level phases (piston case here). All phases can be projected in a unique way on this basis.

Eigen modes in the local only piston case
Fig. 2. Eigen modes in the local only piston case. The top line corresponds to the four modes with the highest eigen values, the bottom line to four of the modes with the lowest eigen values. In this second line, we can recognize discrete versions of some common low-order Zernike polynomials: the two astigmatisms and the tip and tilt. Furthermore, the last modes focus more on the corner segments, that are typically the segments that impact the contrast the least, since they are the most obscured by both the apodizer and the Lyot stop. Conversely, on the top line, we can also see that the segments with the most extreme piston coefficients correspond to the segments hidden by neither the apodizer nor the Lyot stop, and so are the segments that influence the contrast the most. This explains why they have the highest eigen values.

Since these eigen modes form an orthogonal basis, they contribute independently to the contrast. Therefore computing a contrast due to a certain phase is equivalent to summing the contrasts of the projections of this phase on the different eigen modes.

As a consequence, this problem can be inverted: from a fixed target contrast, it is possible to reconstruct the constraints per eigen mode. To do so, we fix the contributed contrast of each mode (the sum of these contributed contrasts has to be equal to the global target contrast). From this contrast per mode and the egein value of each mode, it is possible to compute the constraint on each mode. Fig. 3 illustrates this constraints in the case of a global target contrast of 10^-6, where the constraints on the 35 first modes provide equal contributed contrasts of 10^-6/35, and the constraint on the last mode provide a contrast of 0. The way to read this plot is that, for example, our error phase cannot be higher than 1.6 times the first mode + 1.7 times the second mode + … + 9.5 times the 35th mode.

Contributions on the different piston modes
Fig. 3. Contributions on the different piston modes to reach a final target contrast of 10^-6, in the case where only local pistons on segments deteriorate the contrast.

To conclude, it is extremely to compute the constraints per mode with this method. But even more important, it provides a better understanding of the pupil structure and impact on the contrast, targeting the critical segments. It is then easier to optimize the backplane structure or the edge sensors on these segments to limit their impact on the contrast.

This method of inversion is also applicable to quasi-static stability study and to any other Zernike polynomial.

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