4.4 Formation of the interferometric fringes

In this section, I focus on the light combination and the signal detection.

4.4.1 Beam combination

Figure 4.7: Mariotti's classification (adapted from [Mariotti et al. 1992]).

[Mariotti et al. 1992] tried to classify the different types of beam combination (see Fig. 4.7). They have defined 4 levels of criteria:

• Beam étendue: what is the field accessible by the detector at each telescope? If this field is limited to the diffraction pattern of the telescope, then the interferometer is called single-mode, whereas if the interferometer processes more information than the one in the diffraction pattern it is multi-mode. For example in radio, all interferometers are single-mode.
• Beam direction:

  Figure 4.8: All-in-one coaxial (left) and multiaxial (right) 4-way beam combination schemes.

  
  how the light coming from the different arms are combined? If the beams are combined with beam splitters so that they appear to come from the same direction, then the combination is called co-axial (see Fig. 4.8a) and gives a flat tint on the detector. If the beams appear to come from various direction (see Fig. 4.8b) like in the Young's experiment the beam combination is called multi-axial.
• Combination plane:

  Figure 4.9: Combination planes.

  
  does the beam combination occur in the image plane, conjugated with the sky, or, in the pupil plane, conjugated with the telescope pupils (see Fig. 4.9)?
• Relation between the input and output pupils: what is the interferometric field of view? The answer depends on the relationship between the input / output^4.1 pupil geometry. One can distinguish 3 cases:
  1. Fizeau-Stéphan setup where the input and output pupils are homothetic (both pupil separation and diameters).
  2. Densified pupil used by the hyper-telescopes [Pedretti et al. 2000] where the position of the sub-pupils in the output pupil are scaled to the input ones but the diameters of the sub-pupils are magnified.
  3. Michelson-Pease setup where there is no link between the input pupils and the output pupils.
  The resulting field of view ($FoV$) are sorted by increasing size: $FoV_1 \geq FoV_2 \geq FoV_3$. The Fizeau-Stéphan set-up gives access to a larger field of view but is difficult to implement since the homethetic relation must be conserved during the transit of the object. It would require continuously reconfigurable beam combiners.

The tree that corresponds to this classification (see Fig. 4.7) shows the complexity of beam combination in optical interferometry. However, all but one current interferometers have been designed to be single-mode, GI2T-REGAIN being the only one using the multi-mode beam combination scheme.

4.4.2 Fringe coding and detection

Figure 4.10: Multiaxial beam combination and spatial coding of two-aperture fringes. Left: the OPD changes with the $x$-position on the detector. Center: the fringes appear superposed to the beam shape. Right: fringes ($y$-axis) spectrally dispersed ($x$-axis) in the IR channel of GI2T [Weigelt et al. 2000].

Figure 4.11: Coaxial beam combination and temporal coding of two-aperture fringes. Left: different types of OPD modulation. Center: theoretical signal. Right: signal observed with the IR table at IOTA.

Once the beams have been combined, one still needs to detect the fringes. Since optical detectors have access only to the intensity of the electric field, the signal must be modulated in phase in order to measure both the amplitude and the phase of the visibility. The signal measured from the combination of two arms $A$ and $B$ is deduced from Eq. (4.1):

$$I(\phi_{\rm mod}) = I_A + I_B + 2\sqrt{I_A I_B} \;V_O V_I \; \cos(\phi_O + \phi_{\rm mod}).$$ (4.4)

The goal is to evaluate the complex visibility $V_O \exp (i\phi_O)$ of the object. One needs to modulate $\phi_{\rm mod}$ so that the variation of $I(\phi_{\rm mod})$ in function of $\phi_{\rm mod}$ leads to the amplitude of the visibility. There are mainly two types of fringe coding: the temporal or spatial coding.

In the multiaxial combination scheme, since the beams from the different arms come from different directions, $\phi_{\rm mod}(x) = 2\pi(bx/d)/\lambda$. Therefore analyzing the light at different positions on the detector plane, gives the visibility information (see Fig. 4.10). In the coaxial combination scheme, one introduces a variable optical path length on one arm: $\phi_{\rm mod}(\delta) = 2\pi\delta/\lambda$ (see Fig. 4.11). There exists also other types of coding using the polarization or wavelength dependence of the phase, but they are rarely used.

When combining more than two beams, one has to decide if one uses all-in-one or pairwise beam combination. When the number $N$ of telescopes increases the all-in-one combination is prefered because it involves less optical elements. In a pairwise scheme, all beams must be splitted in $N-1$ beamlets to be combined with the other telescopes. The all-in-one solution is displayed both for co-axial and multi-axial combinations in Fig. 4.8. However, one has to be cautious on the redundancy of the fringe frequencies so that the signals from two different baselines are not mixed together. That is why in the multi-axial combination the sub-pupils are separated by non-redundant separations, and, in co-axial combination the OPD scan frequencies and amplitudes are also not redundant.