- 3.1 Modeling interferometer optics
- 3.2 Computing the static electric field maps in the pupil plane
- 3.3 Computing the time-dependent electric field maps in the pupil plane

3 Computing the dynamic instrument response

For the simulation of the aperture synthesis imaging process of an
interferometer the knowledge of the time-dependent optical transfer
function (*OTF*) is of central importance.
The *OTF* is given by the autocorrelation of the pupil function, i.e. the two-dimensional
distribution of the complex electric field in the interferometer pupil
plane when observing an point source.
The task of computing the time-dependent electric field output of the different interferometer
arms for a given baseline is performed by an optomechanical End-to-End
model (EM).
An EM contains models for structural mechanics, control
system, sensors, actuators, various perturbations (e.g. atmospheric
turbulence, wind), and - as a core part - an optical model which computes
the optical signal flow through the instrument.
For realistic simulation of the control loop sensors the source intensity
in the respective wavelength range has to be taken into account.
Modeling the fringe sensor performance requires the knowledge of
the object visibility for the given baseline.

We have developed a novel optical modeling tool (OM) which is specifically designed to meet the requirements for simulation of ground- and space-based astronomical interferometers (Wilhelm et al. 1998; Wilhelm & Johann 1999). A fast coordinate-free ray tracing algorithm (Redding & Breckenridge 1991) forms the basis of the OM. Diffraction propagation is handled either by numerical approximation of the Rayleigh-Sommerfeld integral or by Fourier-optical methods based on the Fast Fourier Transform (FFT). A Jones matrix formalism keeps track of the vectorial electric field. The OM includes a photometry algorithm which calculates the calibrated power flux through the instrument. In contrast to stochastic ray tracing methods of some commercial optical design programs our method makes use of a triangle grid interconnecting the rays. Each triangle is interpreted as an energy-carrying surface element of a locally planar wavefront. Beam propagation is modeled by a sequence of geometrical and physical-optical propagations. The choice of the adequate propagation model (geometrical/physical optics) for a single propagation step between two optical surfaces is made with respect to the respective "beam geometry'', i.e. the relative sizes of beam diameter, wavefront curvature, optical element aperture and wavelength.

Figures2 and 3 show scaled models of a VLT Unit Telescope and a VLTI optical delay line, both established with the OM.

The observation of a point source is simulated by propagating
a plane wave through each interferometer arm using a sequence of
geometrical and physical-optical propagations.
To simulate a naturally polarized starlight beam the computation is done sequentially for two
uncorrelated, linearly polarized beams with mutually perpendicular
polarization directions ("*s*'' and "*p*'') defined in the plane of the
incident wavefront.
The vectorial electric field
distribution in the pupil plane is computed for each "star polarization'' (*s*, *p*):

where
denotes a position in the pupil plane and the subscripts "1'' and
"2'' correspond to the two interferometer arms.
The function
is the vectorial pupil function
which is used to compute the static *OTF* (see Sect.4).

Figure4 shows the distribution of the electric field amplitude and phase (one Cartesian component) in the exit pupil plane of a single VLTI arm (one interferometric beam). The field maps result from a hybrid propagation model combining geometrical- and physical-optical propagations. As the propagation of the starlight beam is within the near-field regime the beam diameter at the exit pupil does not substantially deviate from the 8 cm geometrical-optical footprint (indicated by the white circle in the amplitude map). 92.8% of the optical power is received within the 8 cm circular footprint for a wavelength of 2m. On the other hand the amplitude pattern shows significant diffraction effects (typical "near-field ripples''). The phase variation across the 8 cm diameter corresponds to approximately one wavelength (phase difference ). Outside the 8 cm circle the slope of the phase map is significantly bigger leading to a rapid oscillation between 0 and .

Up to this point we have described the computation of the static electric
field distributions.
The calculation of the dynamic *OTF*s requires the
knowledge of the time-dependent electric field distributions
and
.
Therefore the OM is integrated into an EM.
We have implemented the OM described above within the VLTI End-to-End Model, developed
at ESO (Denise & Koehler 1998; Wilhelm & Koehler 1998).
It simulates the dynamic response of the VLTI to a point source.
Engineering objectives include the analysis of collective effects of
disturbances and the study of interaction of optics and control loops.
Eventually, the model is planned to be used as a diagnosis tool during the commissioning of the instrument.
Simulated disturbances are wind load on telescopes, atmospheric wavefront
piston and tip/tilt, and seismic noise.
Atmospheric scintillation and high-spatial frequency corrugations
of the wavefront are not taken into account.
Mechanical structure modeling is done off-line using finite element software.
The response of the interferometer to disturbances is then modeled in the
End-to-End model using transfer functions between the disturbances (e.g. wind load)
and optical parameters (e.g. optical path length (OPL)).
The dynamic control environment model uses a "linear optical model''
("small-motion model'').
A linear optical model is represented by
a sensitivity matrix characterizing the dynamic behavior of a set of
optical output parameters in the presence of perturbations acting
on the optical system.
Examples for optical output parameters are the
changes in the position or optical path of a ray or the changes in
the Zernike coefficients representing the optical path distribution
of a wavefront.
The usage of a linear optical model assumes that
the perturbations (translations and rotations) of the optical surfaces
are small enough to ensure proportionality between the changes in
output parameters and the perturbations.

For computation of the dynamic pupil function it is assumed that the
time-dependent electric vector field
in the
exit pupil of an interferometer arm *i* is given by the product of a
static vector field
and a time-dependent phase
factor

The time-dependent "phase error''
arises from
fluctuations of the optical path in the exit pupil with respect to
the static situation.
As the displacements of the optical elements
due to the disturbances and active control are sufficiently small
the resulting phase error can be regarded as polarization-independent,
i.e. the *same* factor exp
is applied to all
Cartesian components of the electric vector field.
The equation
links
the phase error for a given interferometric arm *i*to an "optical path error''
which itself can be
expressed as a Zernike polynomial decomposition using *M* orthogonal
Zernike polynomials
(typical number *M* = 15):

(6) |

The *M* time-dependent coefficients d*Z*_{m} are the deviations of the *M*Zernike coefficients with respect to the static situation.
Within the dynamic simulation the coefficients d*Z*_{m} are computed by
a linear optical model represented by a sensitivity matrix :

(7) |

where d
is a vector of size
holding
the deviations of the *M* Zernike
coefficients, d
is a vector of size
holding the
perturbations (rotation and translation; 6 degrees of freedom per
surface) of the *N* optical surfaces along a beam train and is a
sensitivity matrix.
The computation of the sensitivity matrices for the different
interferometer arms is done in the preprocessing phase of the EM
by sequentially applying small perturbations to the optical surfaces,
propagating the starlight beam and watching the changes in the
Zernike coefficients in the respective exit pupils.

If the displacement of the exit pupils has to be taken into account appropriate sensitivity matrices can be defined analogously.

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