Field of View of the VLTI Mid-Infrared Interferometric
Instrument (MIDI)
O. von der Lühe
Kiepenheuer-Institut
für Sonnenphysik
Schöneckstraße 6-7
79104 Freiburg i. Br.
20. 05. 1998
Table of Contents
1 Introduction
The MIDI instrument observes interferometric fringes with the ESO VLT at
wavelengths in the N and Q bands (8µm ... 24µm). It uses beam
combination in a pupil plane conjugate of two inter-ferometer elements
in a co-axial mode and a synchronous time modulation technique for fringe
detection. This method of beam combination is known to cause field dependent
geometric delay. The purpose of this report is to assess the significance
of this effect and the resulting variation of fringe contrast given the
MIDI design specifications. This variation is compared to the variation
of a design where geometric field delay is fully compensated.
This report is based on the most recent design file of the MIDI Programme
at this date.
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2 Field
Effects of a co-axial beamcombiner
The treatment shown in the following sections is taken from Reference
[1].
2.1
Field Dependent Geometric Delay
A field dependent geometric delay occurs when the optical path length of
the elements in an inter-ferometer are balanced for a given position in
the sky by means of delay lines. We shall assume that this position coincides
with the boresight (telescope on-axis) direction of all elements. A source
which is offset from the on-axis direction by an angle ,
where Da and Dd are
the offsets in right ascension and declination from the boresight, experiences
a geometric delay with respect to the (balanced) on-axis direction of
|
(1)
|
represents the baseline between
interferometer elements with indices i and k, projected onto the celestial
sphere in the direction of the boresight. Arrowed quantities represent
two-dimensional vectors. The optical design of the MIDI instrument does
not compensate for or alter the geometric field delay as other designs
do, it enters the measurement unchanged. Geometric field delay is, as the
name says, a pure geometric effect which is independent from wavelength.
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2.2
Monochromatic visibility
Geometric delay is transformed to optical phase shifts at the monochromatic
wavelength l via
|
(2)
|
where represents the 2-dimensional
fringe frequency measured in line pairs per radian - i. e. the coordinate
in visibility space. Evidently, and
consequently wik depend on the wavelength;
variations of the wavelength due to spectral scanning during detection
of fringes or limited spectral resolution will modify the monochromatic
visibility.
For a fixed, monochromatic wavelength l
the detected fringe visibility Vl ,ik
can be expressed with the Fourier transform of
the sky intensity distribution at
the fringe frequency . Instead
of integrating over the entire celestial sphere, the integral of the Fourier
transform need only be evaluated over the effective field of view; i. e.,
the field over which fringes are actually detected. In the case of the
MIDI this integral is confined to the area F on the sky covered
by a single detector pixel. In general, the pixel should be represented
by a weighting function .
For the sake of simplicity we shall assume that the weighting function
is unity inside F and zero every-where else. Then we have:
|
(3)
|
The term in the demoniator normalizes the visibility to unity at the frequency
origin. The expression in eqn. (3) is the exact representation of the visibility
signal from a very narrow-band source that MIDI would detect in the on-axis
pixel. The geometric delay between neighbouring pixels would cause a corresponding
overall phase shift in the measured visibility signal which can be accounted
for in the data analysis process.
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2.3
Finite spectral bandwidth
In practice, the visibility observed in a MIDI pixel will correspond to
a finite spectral band. Depending on the kind of grism used, the spectral
width covered by a pixel may vary. The exact form of the instrumental spectral
response of a pixel is a function of the optical configuration and the
pixel sensitivity distribution. We shall represent the response function
for the sake of simplicity again by a weighting function which is unity
inside the range and zero
elsewhere. l c is the central wavelength
of the pixel considered, and ,
where Rl is the spectral resolution of
the instrument. We then obtain:
|
(4) |
The observed visibility represents the integral of the monochromatic
visibilities over the effective spectral range, weighted with the spectral
distribution of the source. Obviously, spectral and spatial characteristics
of the source now become intertwined and further treatment dependent on
specific source models.
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3 Field
Effects of MIDI
3.1
VLTI and MIDI specifications
The six baselines between the four 8m Unit Telescopes are considered in
this study only. There are too many Auxiliary telescope stations to consider
them all, and the magnitude of the field effects are comparable to those
of the UTs. Figure 1 shows the location of
the Unit telescopes in a NE coordinate system. Effective baselines depend
on the position of the source. Earth rotation effects are therefore taken
into account for source hour angles in the range of ±
6 hr.
The following table presents the specifications of the MIDI instrument
which are considered relevant for the study.
Item
|
Specification
|
Remarks
|
Central wavelength l
c |
10 µm |
N band only case considered. The relative effects
in the Q band would be about 2 times smaller. |
Spectral resolution Rl |
120.55 with grism
2.5 entire N band |
The entire N band is considered for completeness
only. |
Diffraction limit at MIDI detector l
c feff / D |
75 µm |
Based on the overall effective focal length
of 600 mm |
MIDI detector pixel area |
50 µm |
This corresponds to 0.516 arcsec in the sky. |
Figure 1: Location of Unit Telescopes.
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3.2
Effects for the boresight pixel
We shall first consider the pixels of the MIDI which represent the on-axis
position in the sky. These are all pixels at the position of the VLTI field
of view for which the delay lines compensate the optical path. For the
case of a properly aligned field stop or fiber these will be all the pixels
illuminated through the grism. In the case of an extended field (absence
of a field stop etc.) these will be the pixels covered by the spectrum
of the on-axis position.
The geometric delay across the square area of a MIDI pixel is shown
in Figure 2 for all six baselines and for
a source at a declination of -30° in the
form of an AVI video. The delay remains within ±
10 µm throughout the full range of hour angles except for the baselines
between UT1 and UT3/4 where they are larger for hour angles around the
meridian.
Geometric delay effects occur for the detected visibility with MIDI,
compared to a beamcombiner design which compensates for geometric delay
(Fizeau beam combination) when field and spectral effects mix. For example,
the geometric delay at the edge of the pixel causes a variation in optical
phase of (in radians)
|
(5)
|
across the detected spectral band. The amount of phase variation of course
is proportional to the width of the spectral band. The variation in phase
causes a decrease in fringe contrast by a factor
|
(6)
|
Figure 2: Geometric delay within a 50 µm square MIDI pixel.
The horizontal direction corresponds to right ascension, the vertical direction
to declination. The six panels correspond to the baselines between Unit
Telescopes (from top left to bottom right) 1-2, 1-3, 1-4, 2-3. 2-4, 3-4.
The full scale of the color bar represents the range from -8 µm ...
+8 µm.
Geometric delay - animated images to show all hour angles for various
source declinations:
declination 0 degrees
|
declination -30 degrees
|
declination -60 degrees
|
Figure 3: Difference between the optical phase of the central wavelength
l c and the edge of the spectral
band l c + D
l /2 across a MIDI pixel, expressed in waves.
Further details see Figure 2.
Phase difference - animated images to show all hour angles for various
source declinations:
declination 0 degrees
|
declination -30 degrees
|
declination -60 degrees
|
The variation of optical phase across the MIDI pixel for a source declination
of -30° and for a range of hour angles is
shown in Figure 3. The variation never exceeds
the range of ± 0.01 waves for l
c = 10 µm and for a spectral resolution of 443.
Evidently, on-axis geometric effects are very small for the grism mode
of MIDI.
This is also seen from the contrast loss that a source would suffer
at a given position within the pixel field, shown in Figure 4. This
figure must be interpreted as follows. Consider a position of the delay
lines which balances the optical path difference between telescopes for
a position on the sky which corresponds exactly to the center of the pixel.
A point source elsewhere inside the field covered by the pixel would suffer
a contrast loss due to spectral bandwidth smearing which corresponds to
the value shown in the panel.
Figure 4: Fringe contrast loss factor as a function of source position
within a MIDI pixel. The range of the colour bar represents values between
0.9995 and 1.0000.
Fringe contrast loss - animated images to show all hour angles for
various source declinations:
declination 0 degrees
|
declination -30 degrees
|
declination -60 degrees
|
The source declination has some, but not a large effect on the differential
delay and the fringe contrast losses. The conclusion that the field effects
within a MIDI pixel are negligible for a spectral resolution of 443 is
maintained. Results have been obtained for source declinations of 0 and
-60 degrees, but are not reproduced here.
The picture changes completely should one chose to operate the MIDI
with the full N band, i. e. with the grism removed and an N band filter
inserted. The spectral resolution would now be about 2.5. In this case
the phase differences are two orders of magnitude larger and the fringe
contrast loss factor across a pixel approaches zero. This case is illustrated
in the animated figures Figure 5 and Figure
6. The "field of view" of the MIDI in this case depends on the
specification of a contrast loss limit and would amount to at best a quarter
of a pixel. It is seen that such a field changes size and direction as
the hour angle changes.
Figure 5: Phase difference between center and edge of N band for
a MIDI pixel. The range of the colour bar correspnds to ±
1 wave. See also also Figure 3.
Figure 6: Fringe contrast loss for the full N band for a MIDI pixel.
The colour bar represents contrast loss factors between 0 and 1. See also
also Figure 4.
Wide spectral band - animated images to show phase difference and
fringe contrast loss for all hour angles for a source declination of -30
degrees:
phase difference
|
fringe contrast loss
|
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3.3
Effects between pixels
Whenever MIDI is used with an extended field - i. e., more than just one
pixel representing a position in the sky, there will be a geometric delay
between pixels. The delay effects within an off-axis pixel are similar
to the effects discussed in the previous section and require no further
study. We now consider the inter-pixel field delay effects and their impact
on the MIDI operation.
The unvignetted field of the VLTI in the beamcombiner lab is 2 arcsec
on the sky. Four MIDI pixels can cover this field. For the sake of simplicity
we shall assume that pixels will be off by 0.516 and 1.032 arcsec center-to-center
(i. e., 50 µm and 100 µm on the detector) from the boresight
pixel in the direction of increasing right ascension and declination, respectively.
The effects in the opposite direction will just have the opposite sign.
The graphs on the following pages show the geometric delay between pixels
for each of the six UT-UT baselines. The four curves represent the four
positions in the field. "1 delta" refers to one pixel offset from
the boresight direction to increasing declination, "2 delta" refers
to 2 pixel offset, "alpha" refers to offsets in the direction of
increasing right ascension.
It is seen that the geometric delay amounts to up to several hundred
µm or a few dozen waves. With the spectral resolution of the grism,
such a delay would not affect the measured fringe visibility. However,
there is a geometric phase offset between pixels which needs to be taken
into account if the data are to be referenced properly. This can be done
offline in the data analysis stage and requires a well known detector geometry.
If one wishes to scan the white-light fringe on each pixel, this must be
done sequentially by offsetting the MIDI internal delay lines for each
pixel and detecting synchronously the interferogram.
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4 Conclusions
The following conclusions are drawn from this study:
-
With the spectral resolution provided by the MIDI grism, geometric effects
within the field covered by a single pixel on the detected fringe contrast
are negligible. The visiblity measured in a single pixel are a correct
representation of the visibility of all sources inside that field for each
wavelength.
-
Should it become necessary to reduce the spectral resolution of the MIDI
considerably, the study should be repeated. In particular, geometry effects
on the detected fringe contrast are severe if the full N band is
recorded on a single pixel.
-
Geometric delay causes a deterministic fringe offset between neighbouring
pixels which must be accounted for in a wide field mode.
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5 References
[1] O. von der Lühe and N. Ageorges, Imaging
in Interferometry, in High Angular Resolution in Astrophysics,
NATO ASI Series C Vol. 501, A.-M. Lagrange, D. Mourard, P. Léna
(Eds.), Kluwer (1997), sect. 5 and 6.
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This page was created by O.
von der Lühe on 25. Aug. 1998.