A traditional error budget
aims at quantifying the deterioration of the contrast with the 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 cophasing in term of rms
error.

*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.

*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 35^{th} mode.

*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.