## Spatial Filtering and Visibility

The advantage of using the practical characteristics of single-mode fibers to carry and recombine the light (as opposed to bulk optics), first proposed by [2] with his conceptual FLOAT interferometer, is now well established. Furthermore, in the light of the FLUOR experiment on the IOTA interferometer, which demonstrated the "on-sky" feasibility of such interferometers for the first time [4] showed that making use of single-mode waveguides could also increase the performance of optical interferometry, due to the remarkable properties of spatial filtering, which change the phase fluctuations of the atmospheric turbulent wavefront into intensity fluctuations. Indeed, and as schematically shown by Fig. 2, the effect of the single-mode waveguide is to only propagate in its core the part of the incoming electric-field projected on its first mode. As a result, the wavefront at the output of the fiber is perfectly plane. In the image plane (see Fig. 3), this means that the shape of the signal at the output of the waveguide is fully deterministic (as a matter of fact quasi-Gaussian, e.g. [7]), as opposed to multimode instruments where the short-exposed images present the famous randomly distributed speckle

Fig. 2. Schematic pupil plane view of the effect of single-mode waveguides on the turbulent wavefront: at the entrance of the fiber, the wavefront is corrugated by the atmospheric turbulence + static aberrations of the instrument. At the output, only the projection of the electric field on the first mode of the waveguide has propagated and as a consequence, the wavefront is plane

Fig. 2. Schematic pupil plane view of the effect of single-mode waveguides on the turbulent wavefront: at the entrance of the fiber, the wavefront is corrugated by the atmospheric turbulence + static aberrations of the instrument. At the output, only the projection of the electric field on the first mode of the waveguide has propagated and as a consequence, the wavefront is plane

Fig. 3. Schematic image plane view of the effect of single-mode waveguides on the image. In the multimode case (left), the image is the well-known randomly distributed speckle pattern. But the total number of photons in the image is constant (scintillation neglected). In the single-mode case (right), the image is deterministic (Gaussian-like shape) but the total number of photons depends on the coupling coefficient (p) which varies with the turbulence

Fig. 3. Schematic image plane view of the effect of single-mode waveguides on the image. In the multimode case (left), the image is the well-known randomly distributed speckle pattern. But the total number of photons in the image is constant (scintillation neglected). In the single-mode case (right), the image is deterministic (Gaussian-like shape) but the total number of photons depends on the coupling coefficient (p) which varies with the turbulence pattern. And the fact that at the fiber's output the intensity profile is deterministic, that is calibratable is of great significance to the AMBER data reduction process. Indeed this important information can be used as an a priori, there by improving the performance of signal processing. We will come back to this point in the next chapters.

The obvious trade-off in producing perfectly stable intensity profiles from single-mode waveguide is that only a fraction of the light, namely, the coupling coefficient, is enters the fiber (once again, as opposed to multimode experiments where the number of photons remains constant if we neglect the scintillation; see Fig. 3). This coupling coefficient depends not only on the source's extent [5] but also on the turbulence, more precisely the Strehl ratio [3]. This explains why single-mode interferometers also require telescopes equipped with Adaptive Optics systems, in order to optimize the fraction of flux entering the fiber. Nonetheless, this coupling coefficient property has a strong impact on the estimated visibility. Indeed, this latter will be biased both by geometric (static) and atmospheric (turbulent) effects. To cope with this situation [4] proposed to monitor the coupling coefficient fluctuations in real time due to dedicated photometric outputs and to perform instantaneous photometric calibration. And indeed, he experimentally proved that doing so, single-mode interferometry could achieve visibility measurements with precisions of 1% or lower on compact sources.

Later, from its theoretical work, [6] has finally shown that, in the general case of partial correction by adaptive optics and for a source with a given spatial extent, the measured instantaneous visibility could be expressed in the general case as Vj x TF[O*(a)Lij(a)], that is, the Fourier Transform of the object brilliance distribution multiplied by the instantaneous interferometric antenna lobe, which is turbulent. This easily explains why and how visibility is biased by geometric effect (lobe effect such as in radio-interferometry) and by the turbulence, and why, as one of the consequences, the field of view of single-mode interferometers is limited to one airy disk, that is, © ~ X/D, where D is the diameter of one telescope. This also tells us that strictly speaking, the modal visibility is not the source visibility. However, [15] confirmed from a analytical approach that, in the specific case of compact objects such as dealt by [4], the benefit of single-mode waveguides is substantial, not only in terms of the signal-to-noise ratio of the visibility but also of the robustness of the estimator. From now on, we will consider our observable to be the complex modal visibility.

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