A&A 365, 314-323 (2001)
DOI: 10.1051/0004-6361:20000185
C. Vérinaud1 - F. Cassaing2
Send offprint request: C. Vérinaud
1 - Observatoire de la Côte d'Azur (OCA), Département Fresnel,
2130, route de l'Observatoire,
06460 Saint-Vallier-de-Thiey, France
2 - Office National d'Études et de Recherches
Aérospatiales (ONERA), DOTA, BP 72,
92322 Châtillon Cedex, France
Received 29 August 2000 / Accepted 18 October 2000
Abstract
The general purpose of an adaptive optics system is to
correct for the wavefront corrugations due to atmospheric turbulence. When
applied to a stellar interferometer, care must be taken in the control of the
mean optical path length, commonly called differential piston. This paper
defines a general formalism for the piston control of a deformable mirror in
the linear regime. It is shown that the usual filtering of the piston mode in
the command space is not sufficient, mostly in the case of a bimorph mirror.
Another algorithm is proposed to cancel in the command space the piston
produced in the pupil space. This analysis is confirmed by simulations in the
case of the GI2T interferometer located on Plateau de Calern, France. The
contrast of the interference fringes is severely reduced in the case of a
classical wavefront correction, even in short exposures, but is negligible
with our algorithm, assuming a realistic calibration of the mirror. For this
purpose, a simple concept for the calibration of the piston induced by a
deformable mirror is proposed.
Key words: instrumentation: interferometers - instrumentation: adaptive optics - atmospheric effects - methods: numerical - techniques: interferometric
Author for correspondance: cassaing@onera.fr
For very high spatial resolution, the beams of two or more telescopes can be combined in a stellar interferometer (SI) (Labeyrie 1975). Performance of SIs is limited by the optical path difference (OPD) between the telescopes, induced mainly by atmospheric turbulence. As in AO, this can be corrected using a Fringe Sensing Unit (Cassaing et al. 2000) and a delay line.
Up to now, SIs were limited to the very bright stars because of the severe atmospheric limitations. The main challenge of very high angular resolution is to drastically improve the limiting magnitude. Several SIs under construction plan to use large telescopes corrected by AO techniques, such as the VLTI (Donaldson et al. 2000), the Keck Interferometer (Colavita et al. 1998), and the GI2T (Mourard et al. 1994), for which an AO system based on a curvature WFS and a BDM is under construction (Vérinaud et al. 2000).
When using AO in a stellar interferometer, the control of the
DM piston mode is a critical issue since a piston induced by each DM creates
an additional OPD, shifting the interference fringes. A solution is to
consider all the sensors (WFSs and fringe sensor) and the DM actuators as a whole
(dashed line in Fig. 1), driven by a global control system (Roddier 1999).
![]() |
Figure 1: General layout of a stellar interferometer equipped with AO |
Open with DEXTER |
In this paper, we investigate further the interaction between AO correction and OPD. In Sect. 2, we explain the origin of the DM piston effect and describe a correction method. In Sect. 3, a simulation of the piston effect for the GI2T is presented. In Sect. 4 we propose an experiment to calibrate the piston contribution of each DM mode and evaluate its precision in terms of error propagation on the sky.
For the good understanding of the piston effect, it can be useful to consider
the general control algorithms of an AO system and the behavior of the DM. In
the weak deformations regime, a DM can be characterized by the vectorial basis
of its influence functions
,
defined as the DM phase response at a
point
in the pupil, to a unit voltage applied to each actuator i.
The components
of a given DM phase response
on the influence functions form the command vector
applied
to the mirror. This can be written:
![]() |
(1) |
![]() |
(2) |
For a BDM, the influence functions can easily be computed in the case of an
infinite plate by solving a Poisson equation in the Fourier space
(Kokorowski 1970):
![]() |
Figure 2: Simulated deformations of a BDM. Influence functions of the central electrode (top) and of an electrode of the outer ring (bottom); the dashed line represents the outline of the useful pupil and the spikes denote the position of the 3 fixed points of the DM mount |
Open with DEXTER |
![]() |
Figure 3:
3D view and plot of a mirror section for the piston and defocus modes of a BDM. Top: the piston mode
![]() ![]() ![]() |
Open with DEXTER |
![]() |
(5) |
To compute a command matrix allowing to close the loop without saturation, the
undetected modes are discarded in the command space by setting their huge
gains
in E(-1) to zero and leaving the other
unchanged. The DM piston mode
is
thus never commanded. We may call this approach classical control.
To close the loop and to ensure stability, the command is usually temporally
filtered by a simple integrator with a given gain g. The command vector at
the time ti is given by:
To describe how piston can be introduced, we suppose the mirror is able to
produce a perfect piston mode so that
.
The piston in the pupil is then given by:
![]() |
Figure 4:
Matrix
![]() |
Open with DEXTER |
The simplest strategy for correcting the piston effect is to completely
discard the DM piston induced in the pupil at the command level. Each
influence function contributes to the piston mode in the pupil plane by a
quantity
given by:
The piston correction in the control loop can be obtained by subtracting from
the command vector
the piston command weighted by a coefficient
representing the overall contribution of
to the pupil piston. This
coefficient is simply the scalar product between
and the
vector of components
.
The
command giving a zero-mean deformation may be written:
![]() ![]() ![]() |
(13) |
The main feature of this algorithm is that no extra computation time is needed for the piston effect correction.
The first limitation of the piston correction algorithm is saturation. The
absolute values of the voltages applied to the actuators are limited in
practice to a given maximum
.
Thus, the erratic saturations
of the commands occurring during atmospheric compensation induce phase
correction errors, and also jumps in the average of the DM phase over the
pupil (i.e. of the piston
), because of a mis-correction of the
piston mode. The piston mis-correction
can be written:
Piston residuals due to saturation can be corrected by subtracting
from the applied command when saturation is present. Furthermore
this correction needs to be applied only during the loop iterations where the
command saturates, which occurs only in a sporadic way. Accordingly, the
saturation correction doesn't add any significant temporal delay in the loop.
The saturation issue is illustrated by simulations in Sect. 3.
Another limitation is hysteresis of the piezoelectric materials inducing
departure from the linear hypothesis. For most of the materials used for BDMs,
the amplitude of the hysteresis cycle curve is comprised between 5 and .
The complete study of the hysteresis effect is a difficult task because
of the complexity of the behavior of the mirror as every deformation depends
on the entire trajectory of the commands applied to the mirror. However an
analytical study of the hysteresis effect on the piston mode (Tordo et al. 2000)
shows that the maximum error remains negligible at least for short
time-scales.
To evaluate the bimorph piston effect during atmospheric compensation, the
GI2T with two curvature-based AO systems and their close-loop temporal control
have been simulated. The turbulence spatial and temporal characteristics are
obtained by shifting three phase-screens based on von Karman statistics in
front of the pupils, with an average wind speed of
.
At
each iteration, the DM phase is recorded and projected on the Zernike-like
modes
and on
giving the piston in the pupil.
We first show the effect of classical AO control when the DM piston command
is filtered.
![]() |
Figure 5:
Piston contribution of Zernike-like modes for one BDM (
![]() ![]() ![]() ![]() |
Open with DEXTER |
In order to determine the contribution to the piston of each correction mode
we compute the covariance matrix
of the components
of the
DM phase
on the Zernike-like modes.
![]() |
(18) |
To evaluate the piston effect on fringe
contrast
measurements for a given
exposure time, we assume that the pistons induced by the two DMs are
uncorrelated. The OPD induced by the two DMs, as shown in Fig. 6, is
rather fast compared to the atmospheric OPD based on the model of Conan
(Conan et al. 1995). Indeed the bimorph OPD cut-off frequency
is greater than the atmospheric OPD cut-off
frequency
due to aperture filtering.
![]() |
Figure 6:
Temporal power spectrum of the OPD induced by two bimorph mirrors in
classical control and with the K matrix, compared with the atmospheric OPD.
The conditions are:
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
Such a strong high frequency additional OPD leads to a significant fringe
contrast loss, even in short exposures. The variance
of the OPD during the exposure
time
can be computed from classical spectral theory
(Tango & Twiss 1980):
![]() |
Figure 7: Fringe visibility attenuation versus exposure time due to bimorph piston (solid line) and atmospheric piston (dashed line) in the conditions of Fig. 6 |
Open with DEXTER |
The induced piston can be considerably reduced using the command matrix Kwith saturation effect correction. Assuming a perfectly known
vector the residual OPD becomes negligible
as shown in Fig. 6.
To illustrate the need of saturation effect correction, we analyze the DM
piston residual under strong turbulence. The standard deviation of the residual
piston due to saturation, by using the command matrix K alone, can be very
large. For
it is about
as shown in
Fig. 8. We plotted in the same figure the BDM piston residuals by
using the algorithm of the saturation effect correction defined by
Eq. (15). The standard deviation of these residuals are less than
ten nanometers.
These simulation results show that the piston effect can be very well
corrected if the contribution to the piston of each actuator is known. In
Sect. 4 we propose a method to calibrate experimentally the
vector, which is the key parameter for piston
correction.
![]() |
Figure 8:
Piston residuals due to saturation (with command matrix K) under
strong turbulence (top) and residuals for saturation effect correction
(bottom).
![]() ![]() |
Open with DEXTER |
![]() |
Figure 9: Optical scheme for measuring the piston contribution of the DM correction modes. (DM: deformable mirror, RL: Relay Lenses for pupil transfer) |
Open with DEXTER |
Before starting the calibration, one must first determine the piston command-vector
of
the DM. This vector can be obtained by the SVD of an interaction matrix recorded with the WFS. It
is usually the eigen-vector of matrix X, resulting from the SVD (Sect. 2.1), corresponding to the lowest eigen-value. If the influence functions of the DM are known, from a Zygo interferometer for example, the piston command can also be determined by least-square techniques. In this latter case, the user can also build a more convenient set of mirror modes like for example Zernike-like modes.
The calibration procedure can be performed either in a zonal or in a modal way. In the first case, each actuator is calibrated individually in order to determine directly the
vector. In the second case, a set of new modes orthogonal to the piston in the pupil can be built and injected in the modal command matrix computation.
We show here how to calibrate the piston contribution of Zernike-like
modes of the DM, represented by the change of base matrix Odefined in Sect. 2. If a zonal method is used, the
modes are merely to be replaced in this description by the influence functions
.
The intensity at a given point
in the recombined pupil is given by:
The general process of piston calibration consists in modulating the mirror
mode to characterize at a given temporal frequency
,
with a deformation amplitude less than
.
The
overall intensity fluctuations measured on the single-pixel detector are
recorded and can be processed by synchronous detection.
![]() |
Figure 10:
Simulation of interferograms in the recombined pupil of the coma
Zernike-like DM mode
![]() ![]() |
Open with DEXTER |
When a non zero-mean mode Zernike-like mode is driven with a sine signal for example, the
overall intensity at the output is characterized by fluctuations
(Fig. 11) with a significant contribution at f0. Now this
mode has to be modified by adding the DM piston command
weighted
by a coefficient
to the command producing the
in order to cancel the piston in the pupil. The new deformation to
modulate writes:
![]() |
(22) |
![]() |
(24) |
For sensitivity purposes we wish to apply a modulation with the largest
possible amplitude. With a maximum deformation amplitude of ,
the
response already reached the non-linear region, so that some intensity
fluctuation remains even when the driven mode is zero-mean as shown in
Fig. 12. However these results show that calibrating
Zernike-like modes, which have a more symmetric shape than single influence
functions, may induce negligible errors except for a few modes.
![]() |
Figure 11:
Simulated power spectrum of intensity fluctuations for the coma mode
driven with an amplitude of ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
![]() |
Figure 12:
Simulation of piston errors due to non-linear
effects: intensity fluctuation transposed to piston error in nm for
piston-free modes driven with a
![]() ![]() |
Open with DEXTER |
![]() |
Figure 13:
Residual piston error of bimorph mirror during atmospheric
compensation versus mode number for a
![]() ![]() ![]() ![]() |
Open with DEXTER |
For calibration purposes the science beam of an interferometer is sometimes
spatially filtered by single-mode optical devices (Coudé du Foresto et al. 1998)
(Malbet et al. 1999). This leads to an apodization of the intensity in the pupil,
defined by the function
of Eq. (3) close to a
truncated Gaussian (Ruilier & Cassaing 2000). Thus the method presented in
Sect. 4 can not be used just as it is. Indeed the same spatial
filtering as in the science beam must be applied in a focal plane of the
calibration bench. In this case the calibration can be done only with a square
signal modulation because the intensity after spatial filtering depends on the
wavefront shape. If a sinusoidal modulation is used one should use the free
interferometric output of the calibration bench (output 2 of
Fig. 9). The signal to be recorded is then the normalized
difference of the overall intensity between the two arms.
A second order effect of single-mode filtering could be the difference between the shapes of the fundamental mode for different bandwidths. The correction modes should then be optimized to minimize the piston induced by the bimorph mirrors in the spectral bands of observation.
As instrumental characteristics may significantly affect the fringe contrast measurements of an astrophysical object, the AO piston calibration processes we considered could greatly benefit from direct measurements through the science instrument (Fig. 1) in order to include all the effects.
The effect of AO correction on the fringe contrast measured with a Michelson Stellar Interferometer has been analyzed. Control of the deformable mirror piston mode is critical since it is not seen by the WFS, but can induce contrast losses. Our analysis shows that filtering the piston mode in the command space for a bimorph mirror is not applicable, since this leads to severe contrast losses in typical conditions, even for short exposures. By deriving the general formalism of piston control in AO, we demonstrate that the fast piston motions can be corrected without any significant additional computation time. However, this requires a prior knowledge of the absolute influence functions of the deformable mirror, including the piston term.
In this goal, we propose a calibration bench able to measure the piston contributions; this set-up could be used to qualify the performances of piston control and could also be upgraded to a built-in facility for on-site calibrations.
Acknowledgements
The simulations of the GI2T AO system presented in this paper are based on F. Rigaut's AO simulation package. The authors are very grateful to him. We also wish to thank P.-Y. Madec (ONERA) and A. Blazit (OCA) for very fruitful discussions and we acknowledge heartily the anonymous referee for very constructive suggestions. This work is supported by the Technology and Research department of the region Provence-Alpes-Côte d'Azur and by the CNRS (France).