M2

Fig. 20. A schematic representation of the Michelson stellar interferometer. Two sources are represented in the figure to compare the chief rays' behavior for an on-axis object and an off-axis object between the two PSFs should be D/b. The PSF of this interferometric setup is reported in Fig. 21. The left PSF is achieved for an on-axis object, and the center PSF is achieved for an off-axis object. In both cases, a noticeable gain is achieved in angular resolution because of the small fringe spacings in the PSF. As anticipated, the fringe spacing is a factor D/b smaller then the PSF of the single lens of diameter D (left plot of Fig. 21). It is usual, however, for the PSF shape to change with field of view positions. This means that an image composed of several points will be reimaged through the mentioned system using a different PSF per each point of the FoV. In this case, the image we obtain is no longer a good reproduction of the observed object intensity pattern. Why did this problem not occur with the lens or with the lens and two sub-aperture we described before? As already mentioned, in the lens case the lens glass compensates for the geometrical difference in optical path so that all the rays from a given source arrive in phase at the focal point. In this case, something different happens. A simple idea of the system behavior is found by considering the small inset B in Fig. 20. The two wavefront portions, coming from an off-axis object and sampled by the two mirrors, arrive on the lens with an optical path difference highlighted in the inset by the solid arrow. This is the optical path difference OPD 2, which the big lens L2 would have compensated for, to keep the PSF stable in the field of view. However, the optical path that the lens L1 compensates (at that particular position in the field of view) is different from the one considered above and corresponds to the optical path difference OPD 1 identified by the dashed line.8 For this reason, the two chief rays from the considered mirrors do not interfere positively in the case of an off-axis source. In other words, the sinusoidal pattern we introduced in Fig. 19 is shifted, with respect to the mirrors PSFs, because of the uncompensated optical path difference. This effect changes the PSF shape with respect to the on-axis case. It is easy to see that this effect is

Fig. 21. Left: the on-axis interferometric PSF. The PSF maximum is in the center of the profile. Center: the off-axis PSF. The maximum of the PSF is not in the center of the profile. Right: the PSF of the single aperture of diameter D

8 The dashed line is obtained by considering the wavefront surface perpendicular to the initial propagation direction, identified by the off-axis object position in the sky.

not present on the on-axis object, where for symmetry reasons the sinusoidal pattern remains centered on the mirrors PSF.

Now, what is the difference of the present setup from the lens and the two sub-apertures case we already considered? In this interferometric case, the entrance pupil (the two mirrors of diameter D placed at distance b) is not purely scaled on the re-imaging lens L1. In fact, the mirror diameter remains the same while the mirror separation is scaled from b to b'. This violation to a pure scaling or re-imaging of the entrance pupil of the interferometer generates the optical behavior, where the PSF is not constant with the field of view. An interferometer where the pupil is purely scaled in all the optical train is called homothetic and creates real images of the source. However, no scaling is perfect, and so it is important to understand what is the limit that can be tolerated, when an interferometer is used to produce real images. Let us consider the left picture of Fig. 22, where a homotethic configuration is sketched. Note that the two lenses have been added before the folding mirrors to collimate and compress the beams. In formulae requiring the homotheticity means using the symbols defined in figure:

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