and
K
0
ij
can b e computed from the
K
ij
. There is an innite number of p ossibilities for choosing a set
of mirror mo des and any of them could actually b e used. However, since the deformations achieved
by continuous deformable mirrors are mostly low frequency functions, usually some of the rst Zernike
polynomials can be well approximated. It is then convenient to choose these Zernike-like deformations as
mirror modes so that equation 5 can b e directly used to estimate their temp oral cut-o frequency. The
remaining mo des can be chosen arbitrarily and an upp er b ound for their cut-o frequency is given by
the highest Zernike polynomial with a signicant contribution in their decomp osition. In section 4, we
will detail the choice of such a particular base for PUEO. For now, we just dene
f
M
as the maximal
temporal cut-o frequency of the mirror mo des in the turbulent phase.
2.3 Measurement of the residual phase
After correction, the wave-front is splitted: one part is used to actually form the corrected image onto
the detector; the other part is directed onto a wave-front sensor (WFS), so that the residual phase can
be measured. The WFS is a device able to sample the derivatives of the corrected phase
"
(
x
; t
), at
a nite numb er of p oints (sub-ap ertures) on the pupil. Commonly used WFS include Shack-Hartmann
wave-front sensors and curvature wave-front sensors [15]. In any case, the WFS measurement is a
n
-
dimensional vector, where
n
is the number of sub-ap ertures in the case of the curvature WFS and twice
this number in the case of a Shack-Hartmann WFS. We denote
W
(
"
(
x
; t
)) the WFS measurement of
"
(
x
; t
),
in the absence of any measurement error
.
The WFS is usually supp osed to work in its linear domain so that
W
can b e considered as a linear
operator. By applying some known voltages to the DM in the absence of turbulence and recording the
corresponding WFS measurements,
W
can b e calibrated on the mirror space
M
. From this calibration,
the WFS response to the mirror mo des can be deduced and expressed as the columns of a
n
m
matrix
D
, called
modal interaction matrix
. Usually, the number of sub-ap ertures on the WFS and the numb er
of actuators on the DM are dimensioned so that
n
m
. Using equations 8 and 12 we can write:
W
(
"
(
x
; t
)) =
W
(
"
k
(
x
; t
)) +
W
(
?
(
x
; t
)) =
D
"
(
t
) +
W
(
?
(
x
; t
)) (14)
Another way to calibrate
W
is to use a physical mo del to simulate the WFS. In reference [16] for instance,
Shack Hartmann and curvature WFS were simulated in order to compare their resp ective performances
in an AO system. These simulation co des allow to compute the WFS resp onse for any phase function
of
E
(i.e. not only of
M
as above). As a test of the accuracy of the WFS model, a simulated modal
interaction matrix can b e computed and compared to the modal interaction matrix measured as ab ove.
For the curvature WFS used in PUEO, a very go od agreement was found [17].
All the wave-front sensing techniques entail detecting photons coming from a guide star during a very short
integration time (WFS integration time, typically 1 to 50 ms). The WFS measurements are therefore
aected by detection noise which results from the statistical uctuations of the incoming photon ux
(photon noise) as well as the imp erfections of the photon detector (dark current, read-out noise...). The
eect of all these undesirable contributions to the WFS measurements may b e summarized as an additive
random vector
n
w
(
t
): the
measurement error
. The actual WFS measurement
w
(
t
) can then b e expressed
as:
w
(
t
) =
D
"
(
t
) +
W
(
?
(
x
; t
)) +
n
w
(
t
) (15)
2.4 Reconstruction of the residual phase
In an AO system, it is the task of a real-time computer (RTC) to up date the shap e of the deformable
mirror so that the spatial variance of the residual phase remains minimal. However, in order to compute
the actual commands to be applied to the DM, the RTC needs rst to derive an estimate of
"
k
(
x
; t
),
the mirror component of the residual phase, based on the WFS measurements. The modal approach
[18] has proved to be very ecient to solve this problem and is therefore used in many current AO
systems, including PUEO. This technique allows to identify the directions (mo des) of
M
which give a
zero measurement on the WFS. These invisible mo des depend on the geometry of b oth the DM and the
WFS and extensive design studies are usually performed to reduce their numbers. They always include
the piston mode, which has no eect on the image formation, and sometimes some other modes. The
invisible mo des must be excluded from the correction loop so that the actual phase comp ensation is not
4
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