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

2 Field Effects of a co-axial beamcombiner

2.1 Field Dependent Geometric Delay

2.2 Monochromatic visibility

2.3 Finite spectral bandwidth

3 Field Effects of MIDI

3.1 VLTI and MIDI specifications

3.2 Effects for the boresight pixel

3.3 Effects between pixels

4 Conclusions

5 References

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 f_eff / 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. Go back to Table of Contents.

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 - animated images to show all hour angles for various source declinations:

declination 0 degrees

declination -30 degrees

declination -60 degrees

Figure 3:

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 - 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:

Figure 3

Figure 6:

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

Imaging in Interferometry

High Angular Resolution in Astrophysics,

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This page was created by O. von der Lühe on 25. Aug. 1998.

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