- 4.1 Building the optical transfer function
- 4.2 Multiaxial ("image plane'') beam combination
- 4.3 Coaxial ("pupil plane'') beam combination
- 4.4 Taking temporal coherence into account
- 4.5 Simulating polarized objects

4 The time dependent point spread function of the interferometer

4.1 Building the optical transfer function

This section describes the computation of the point spread function (PSF). As the PSF is the image of a point source the function in Eq.(1) becomes a -function corresponding to a constant spectrum .

The spacing of the fringes that one obtains with an interferometer depends only on
the shape of the exit pupil, or better, its autocorrelation, the *OTF*.
In Fig.5 we show three different one dimensional pupil
functions and their corresponding MTFs.
The first two pupil functions show multiaxial beam combination,
the last coaxial beam combination.
If the pupils are so close together that the peaks in the *OTF* are overlapping,
the information in the interferometric peaks cannot be fully retrieved.
A loss of information in the interferometric peaks means a limit in the
interferometric field-of-view.
This can be understood if one remembers that an extended object leads to
variations on small scales in the object complex visibility, which are not
retrievable anymore in the entangled peaks.
Using coaxial beam combination, one cannot disentangle the interferometric
signal from the photometric signal anymore, since all peaks fall on
top of each other (see Sect.4.3).

In our approach we construct the pupil function and then calculate the *OTF*from the output of the interferometer simulations described in
Sect.3.
Since we must have access to all parts of the *OTF* separately without mixing them,
we are internally computing the *OTF* always with well separated pupils
and rearrange the peaks later to match the real *OTF*.
A simulated two dimensional MTF for image plane beam combination with two telescopes
and well separated exit pupils can be seen in Fig.6.

Figure 6:
Modulation transfer function (MTF) for a multiaxial beam combiner.
The structure of the MTF is mainly given by the autocorrelation of two uniform disks |

The outputs from the optomechanical model are static electric field maps, dynamic
Zernike coefficients for the time-dependent OPL maps,
and the dynamic lateral pupil positions.
For each of the two input polarizations there are three static electric field maps,
corresponding to the three Cartesian components of the electric field^{}.
To reconstruct the exit pupil function from
this output for a certain point in time, we take the static electric
field maps and multiply them with the dynamic phasors
exp
(see Eq.5).
To simulate the combination of beams of different sizes (e.g. VLTI beams related to UT and AT) we use a virtual beam
expander/compressor adjusting the beams to the same diameter.

From the vectorial pupil function
(Eq.4)
one gets the internally used *OTF*^{Pol}by computing its autocorrelation (Goodman 1968):

where
is a two-dimensional, dimensionless vector in the spatial
frequency domain (
). Within the model, the calculation of
the *OTF* is performed using two Fourier transforms:

(9) |

The simulation of the imaging process as described in Sect.2 is done independently for the two star polarizations () using the two transfer functions and .

This way we construct an *OTF*_{i} for every timestep *t*_{i}
of the dynamic simulation.
The length of a single simulation interval
is chosen to ensure stationarity of the atmospheric perturbations.
Over an exposure time *T* we have typically a few *OTF*s.
It is the change of the *OTF* with time, which is degrading
the interferometric signal, and in which we are interested.
One gets the *OTF* for the whole exposure time when averaging
the *OTF*_{i}.

Since the dimension of the electric field is
,
the
autocorrelation above has the dimension
.
One retrieves intensities in
when multiplying by
with
and
being
the dielectric constant and the permeability in vacuum.
From here one comes
to energies (and finally photons) by multiplying with the area of the beam *A*and the length in time of the single simulation interval .

The *OTF* constructed in this way is not dimensionless as
described in Eq.(8),
but already holds the photometric information.
The object complex visibility we are multiplying with this *OTF* thus
has to be dimensionless and normalized to one.
Since we are using one object complex visibility for each polarization, the sum of these
two visibilities has to be normalized.
For an unpolarized object these two visibilities are identical and normalized to 0.5.

Since the *OTF*s already hold the intensity information, one can
directly read the correlated flux for a point source from the *OTF*.
It is the sum of the maxima of the interferometric peaks (or twice the
height of one peak).
The total flux is the maximum of the photometric peak,
the uncorrelated flux is then the difference between the total flux and
the correlated flux.
It should be kept in mind that the two interferometric peaks are symmetric and
carry exactly the same information.
An extension of the described method to the case of
extended objects is given in Sect.5.

The combination of more than two beams is performed by using the same method as
for two beams.
To compute the *OTF*, the *N* beams have to be arranged in a non-redundant way allowing
to retrieve the photometric peak and the *N*(*N*-1)/2 interferometric peaks.
Afterwards one has to rearrange them according to the exit pupil.

4.2 Multiaxial ("image plane'') beam combination

For combining the beams in the image plane the exit pupils of two or more interferometer arms are aligned side by side in the pupil plane (see Fig.7). Imaging this plane by an optical subsystem results in a spatial fringe pattern. The exit pupils have to be aligned so there is no redundancy in the distances between them. Usually the exit pupils are set on a line, with non redundant spacing, leaving the perpendicular axis available for spectroscopy (see Fig.8). A more compact setup can be achieved by using two dimensions, taking for example a trapezium structure with four exit pupils (Fig.9).

Figure 8:
Non-redundant pupil function and the corresponding modulation transfer
function for four beams and a linear setup |

Figure 9:
Non-redundant pupil function and the corresponding modulation transfer
function for four beams and a compact setup |

When observing in multiaxial mode one frame per atmospheric coherence time is taken.
A usual method for estimation of the visibility amplitude *V* for an interferometer
combining two beams,
which have been spatially filtered to the extent of a single aperture Airy disk,
is given in Roddier & Léna (1984):
By taking the Fourier transform and squaring it one computes the power spectrum
(squared MTF) of the spatial fringe pattern.
The power spectrum has non-overlapping low and high-spatial
frequency terms (the single photometric peak and the two interferometric peaks).
The ratio of the high to low frequency energies gives a good estimation of the term
.

For observations with a field-of-view exceeding the size of an Airy disk, one has to evaluate the values at every single spatial frequency within the interferometric peaks of the power spectrum.

This mode will be used by the AMBER beam combiner for the VLTI (Petrov et al. 1998). In Fig.10 a simulated image plane combined interferogram can be seen.

4.3 Coaxial ("pupil plane'') beam combination

Most interferometers are combining the beams coming from the arms of the interferometer in the pupil plane. This means that the exit pupils are aligned pairwise on top of each other. Afterwards usually an optical subsystem focuses the pupil on a detector (see Fig.11).

To measure the fringe, the observer has to step through the fringe in time,
namely modulating the optical path in one of the interferometer arms.
One way to achieve this is to take
four frames per atmospheric coherence time, each of which has
(0, 1, 2, 3)
optical path difference (OPD) added to one beam.
The visibility is estimated by determining the energy in each of the
four frames (
*A*, *B*, *C*, *D*) and computing the modulus of
the object complex visibility according to:
.
The phase of the object complex visibility can also be computed
from the four energy bins
*A*, *B*, *C*, *D* (Shao & Staelin 1977).

For the beam combination in the model this means that
one has to add 0, ,
,
to the phase in one of the interferometric *OTF* peaks, and subtract the same value in the other
interferometric peak.
Figures12 and 13 illustrate what consequence this
additional phase has on the *OTF*.

Figure 13:
Same as Fig.12, but for a coaxial beam combination.
In the upper left the two interferometric peaks are enhancing the
height of the photometric peak (no phase shift), in the two right
drawings the interferometric peaks cancel, leaving only the
photometric peak (phase shifts
and ),
while in the lower left the two interferometric peaks cancel the
photometric peak, leaving no signal (phase shift ) |

This beam combination mode is foreseen with the MIDI instrument for VLTI (Leinert & Graser 1998). Four Airy disks of varying intensities, which form a typical interferogram can be seen in Fig.14.

When combining more than two beams in coaxial beam combination
it is necessary to sample the fringe with more than four steps in time.
This leads to different steps in phase for the interferometric peaks, which are now
rotating around the imaginary axis with different speed.
One of the beams is not delayed at all as another beam has to be delayed
by steps of
,
just like in the case of two beam combination.
The third beam has to be delayed with steps of
.
A total of 12 exposures has to be taken.
For the six interferometric peaks of the *OTF* follows that two have to
rotate by steps of ,
two with ,
and the last two with .
The first make three rotations during one fringe acquisition, the second
two rotations, and the latter one rotation.

4.4 Taking temporal coherence into account

Within the modeling software we assume
the validity of the quasi monochromatic approximation.
A finite bandwidth of the source which is much smaller than the center
wavelength/frequency can however be taken into consideration.
This leads to a degradation of the visibility.
In the picture of correlated versus uncorrelated flux this means that
the correlated flux becomes less.
Thus one has to lower the interferometric
peaks in the *OTF* by a generally complex factor
.
is the degree of temporal coherence.
It is a function of the OPD between the two interfering wavefronts and is
computed as the Fourier transform of the power spectral density of the source
(Goodman 1985).
As a first approach we average the OPD between the beams across their
areas and multiply the interferometric peaks in the *OTF* with this value.
If the average OPD substantially exceeds the coherence length of the starlight the
degradation factor
becomes zero and the fringes disappear.
This computing approach does not consider a variation of the OPD across the beam areas.

4.5 Simulating polarized objects

A polarized object can be characterized in every pixel of the intensity distribution
by three quantities:
the flux, the degree of polarization, and the direction of polarization.
In the model the polarized and the unpolarized flux are separated.
The polarized flux is projected on the two polarization axes used within
the optomechanical model.
The unpolarized flux of the object is divided by two and added to the two
polarization maps.
From these two maps we calculate two object complex visibilities.
Both are multiplied with their corresponding *OTF*s.
Now the two different visibilities have to be added to retrieve the measured signal.

Copyright The European Southern Observatory (ESO)