Issue 
A&A
Volume 625, May 2019



Article Number  A48  
Number of page(s)  9  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201834013  
Published online  08 May 2019 
Singlemode interferometric field of view in partial turbulence correction
Application to the observation of the environment of Sgr A* with GRAVITY
^{1}
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
email: guy.perrin@obspm.fr
^{2}
European Southern Observatory, KarlSchwarzschildStr. 2, 85748 Garching bei München, Germany
email: jwoillez@eso.org
Received:
3
August
2018
Accepted:
13
March
2019
Context. The GRAVITY instrument on the ESO VLTI is setting a new mark in the landscape of optical interferometers. Long exposures are possible for the first time in this wavelength domain, delivering a dramatic improvement for astrophysics. In particular, faint objects can be observed at the angular resolution of the VLTI, with exposures exceeding by many orders of magnitude the coherence time of atmospheric turbulence.
Aims. Modern interferometers, especially those that combine light collected by large telescopes such as the Unit Telescopes of the VLT, benefit from partial correction of atmospheric turbulence. We investigate in this paper the influence of atmospheric turbulence on the maximum field of view of interferometers such as GRAVITY, where wavefronts are filtered with singlemode fibres. The basic question is whether the maximum field of view is restricted to the diffraction limit of single apertures or if it is wider in practice. We discuss in particular the case of the field around Sgr A* , with an emphasis on the detectability of the massive mainsequence star S2.
Methods. We theoretically investigated the interferometric and photometric lobes of the interferometer and took atmospheric turbulence into account. We computed the lobe functions with and without partial correction for atmospheric turbulence.
Results. The main result of the paper is that the field of view of the interferometer is widened by tiptilt residues if higher orders of atmospheric turbulence are corrected for. As a particular case, the S2 star can be detected in interferometric frames centred on Sgr A* even when the distance between the two objects is up to about twice the diffraction limit of a single aperture. We also show that the visibilities are not biased in this case if the pointing accuracies of the fibres are of the order of 10 mas.
Key words: instrumentation: interferometers / instrumentation: high angular resolution / atmospheric effects / Galaxy: center / instrumentation: adaptive optics
© G. Perrin and J. Woillez 2019
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The question of the field of view in opticalinfrared interferometry is still a matter of investigation. It was long thought that the question of the field of view of an interferometer was connected to the pupil configuration. Traub (1986) stated the golden rule of interferometry: the maximum field of view would be preserved in an interferometer if the entrance and exit pupils were homothetic because in this case the interference pattern is the result of the convolution of the object by the interferometer response. Labeyrie (1996) proposed the hypertelesope concept, where the exit pupil is densified relative to the entrance pupil. The convolution relation is lost in this case, as is also the case for most interferometers. The exit pupil is always densified in practice for reasons of the signaltonoise ratio when limited by detector noise because this minimises the number of fringes across the diffraction pattern. Lardière et al. (2007) showed that even in the case of pupil densification, the field of view does not so much depend on the exit pupil, but is given by the smallest separation in the array s: λ/s. Direct imaging can be achieved by a hypertelescope in a relatively large field of view.
Beyond array and beam combiner geometries, turbulence also plays a role in interferometric imaging field of view. Roddier & Lena (1984a,b) investigated the effect of turbulence in both the singlespeckle and multispeckle interferometry cases for objects that are not resolved by a single aperture. In particular, they derived the expression of the coherence term measured by the Fizeautype (multiaxial) interferometer. Mourard et al. (1994) later refined this estimator to improve the stability and fidelity of the measured visibilities. Despite these achievements, visibility accuracies reached a few percent to a percent at best. Following earlier ideas to use singlemode fibres to overcome the difficulty of calibrating visibilities in atmospheric turbulence conditions (Froehly 1981), the FLUOR concept was proposed (Coudé du Foresto & Ridgway 1992). Coudé du Foresto et al. (2000) investigated the coupling of starlight to a singlemode fibre at the focus of a telescope under partial correction for wavefront turbulence with adaptive optics. For maximum coupling to the fibre without aberrations, the mode of the fibre has to match the diffraction limit λ/D of each aperture. In presence of turbulence, the injection efficiency is proportional to the Strehl ratio or coherent energy, as demonstrated with simulations by Haubois (2009). Guyon (2002) investigated the effect of turbulence in the shortexposure case on the field of view of a singlemode interferometer: the interferometer measures the object that is apodised by the fibre mode. He showed that calibration over the whole field of view would be difficult, however. In parallel, Mège (2002) built a theoretical framework for the measurement of coherence with singlemode interferometers. He investigated the field of view of a singlemode interferometer and showed that it is limited by the fibre lobe in absence of turbulence and static aberrations or with turbulence in the longexposure case but without correction of turbulence. Most of these findings (without turbulence) were known from radio astronomy, where light is also filtered by singlemode waveguides (Thompson & Moran 2017).
We take the next step in these studies and investigate in this paper the influence of partial correction for turbulent wavefronts on the field of view of a singlemode interferometer. We first describe the field of view that is transported by a singlemode waveguide in Sect. 2. In Sect. 3 we derive the expression of the instantaneous visibility measured by a singlemode interferometer. This expression is generalised to the longexposure case without any phase correction in Sect. 4. It is further discussed in Sect. 5, when a partial correction for atmospheric turbulence is applied, with the consequence on the field of view of the interferometer. The overall results are discussed in Sect. 6
2. Field of view transported by a singlemode waveguide
The coupling of the light collected by a single telescope into a singlemode fibre is set by the overlap integral η between the field at the telescope focus E_{tel} and the fibre mode E_{fib} (Neumann 1988). The wave E( β) injected into the fibre then writes
where β is the local coordinates against which the fibre mode E_{fib} is expressed, and assuming a normalised fibre mode profile, ∫E_{fib}( β)^{2} dβ = 1. When the field at the telescope focus changes, only the global amplitude of the injected wave changes, not the profile E_{fib}( β) of the fibre field, also called carrier wave. This is the mathematical expression of the spatial filtering properties of a singlemode fibre. Through the Parseval–Plancherel theorem, the overlap integral can be converted into pupil plane as follows:
where ℱ^{−1}[] denotes the inverse Fourier transform. The inverse Fourier transform of the focused wave, ℱ^{−1}[E_{tel}], is the complex telescope pupil P multiplied by the wave W collected by the telescope (with W = ℱ^{−1}[E_{obj}], where E_{obj} is the wave from the object in image plane). The inverse Fourier transform of the quasiGaussian fibre mode, ℱ^{−1}[E_{fib}], acts as a quasiGaussian apodising function in pupil plane. Defining P_{◎}, the complex pupil apodised by the fibre mode,
the overlap integral becomes
an average of the collected wave weighted by the complex apodised pupil.
The intensity associated with the coupled field relates to the overlap integral as follows:
where ⟨⟩ denotes the ensemble average operator on the oscillations of the wave W. At optical and infrared wavelengths, the variation timescale of the apodised complex pupil functions P_{◎} as a result of atmospheric turbulence or vibrations, for instance, is much longer than the period of the waves. The ensemble average can be approximated with good accuracy by an average over many cycles of the wave, while keeping the averaging time shorter than the coherence time of the apodised complex pupil functions fluctuations. Doing so, the intensity becomes
Recognising the complex visibility of the object V(u − v) in the term ⟨W(u)W^{⋆}(v)⟩, assuming the integral of the object intensity is normalised: ∫O(α) dα = 1, where O(α) = E_{obj}^{2}(α), and applying a w = v − u coordinates change, the previous equation can be rewritten as follows:
where P_{◎} ⊗ P_{◎} is the autocorrelation of the apodised pupil. With the PersevalPlancherel theorem used once more, the equation above can be converted into object space,
where the inverse Fourier transform of the visibility was identified as the observed object O.
From the equation above, the inverse Fourier transform of the apodised pupil autocorrelation
is recognised as the singlemode telescope lobe L: the map of the intensity coupled to the fibre as a function of offaxis distance. This is equivalent to the point spread function of incoherent Fraunhofer imaging. Finally, to prepare a comparison with the singlemode coherence measurement of the next section, the intensity coupled into the fibre can also be presented as
the zerofrequency visibility of the object apodised by the singlemode telescope lobe.
This result is similar to the wellknown effect of the antenna lobe of radio telescopes (Thompson & Moran 2017) and was also established by Mège (2002). The apodised pupil P_{◎}, just like its nonapodised counter part, is a complex quantity. Its nonzero phase is the result of static aberrations and dynamic turbulence, or the residual phase if an adaptive optics system is present on the telescope. When the strength of the residual turbulence increases, the autocorrelation of the complex apodised pupil narrows and the telescope lobe widens. Then, the coupled intensity becomes more contaminated by surrounding objects when the turbulence is strong. An image of the coupling map can be produced by moving the fibre in the image plane or by stirring a mirror in pupil plane to scan the α directions. This is how the telescope lobes illustrated in Fig. 1 were obtained with the ‘OHANA experiment on the Keck telescopes (Perrin et al. 2006); it is affected by static aberrations and residual atmospheric turbulence.
Fig. 1. Telescope lobe measured on an internal source (left) and on a star with partial adaptive optics correction (right), obtained with the ‘OHANA experiment (Perrin et al. 2006) on one of the telescopes of the Keck Interferometer (Colavita et al. 2013). 
3. Instantaneous visibility measured by singlemode interferometers
We consider E_{i}( β), the wave injected in the fibre at the focus of telescope i. The fields E_{1}( β) and E_{2}( β) from two telescopes overlap in the waveguide after beam combination (e.g. with an Xcoupler, Jocou et al. 2012), with the resulting field
where for the sake of simplicity, the complex coefficients of the beam combiner were chosen equal to unity. The output of the waveguide is conjugated with the detector in a focal plane where the intensity I_{12} is averaged over β:
where ⟨⟩ denotes the same ensemble average operator as in Sect. 2. Developing the square leads to the definition of the complex coherence factor γ_{12} measured between singlemode telescopes 1 and 2, which relates to the overlap integrals η_{1} and η_{2} for each telescope as follows:
The equation above is the twotelescope equivalent of Eq. (5). The same transformations as in the previous section yields a twotelescope equivalent of Eq. (7), linking the complex coherence factor to the object visibility:
where both apodised pupils P_{◎, 1} and P_{◎, 2} are centred on the origin, and B represent the baseline vector joining the two pupil centres. Through the Parseval–Plancherel theorem, the complex coherence factor expressed in object space writes
By defining the singlemode interferometric lobe L_{12} as the inverse Fourier transform of the apodised pupil crosscorrelation,
the complex coherence factor writes
The coherence measured by the singlemode interferometer is also the visibility function of the object that is apodised by the instantaneous interferometric lobe. Mège (2002) obtained a similar result. In the absence of turbulence, this expression is valid regardless of exposure time, as long as it is much longer than the period of the wave. In this particular case, in the absence of any static or turbulent aberration and with identical telescopes, all three apodised pupils P_{◎, 1}, P_{◎, 2}, and P_{◎} are identical, and so are the lobes L_{12}(α) = L(α). The maximum field of view of the interferometer is then set by the pupils that are apodised by the mode of the fibres.
Table 1 summarizes the key definitions related to singlemode flux (Sect. 2) and coherence (Sect. 3). It highlights the similarities between the two sets of quantities and shows that flux appears to be nothing more than coherence at zero spatial frequency.
Summary of the flux and coherence quantities measured with singlemode telescope arrays.
4. Singlemode fully turbulent long exposure
In this section, wavefronts at the telescopes are affected by atmospheric turbulence with Gaussian statistics, and no adaptive optics or fringe tracking systems are used to correct the phase. The expression of the pupil of telescope i apodized by the fibre mode in Eq. (3) can be adjusted to reflect the effect of the turbulent phase ϕ_{i} as follows:
where Π_{◎} is the aberrationfree pupil apodised by the fibre mode, which is assumed identical for all telescopes. Injecting this definition in the expressions of the photometric lobe (L(α), Eq. (9)) and interferometric lobe (L_{12}(α), Eq. (16)), and taking a longexposure average ⟨⟩_{long} on the turbulent phase ϕ, we obtain
where D_{ϕ}(u) is the structure function of the turbulent phase as defined in Roddier (1981). The correlations of the phase within the telescope pupil, set by the phase structure function, are responsible for the widening of the singlemode photometric lobe. We have made two strong assumptions to derive these results. First, the integration time is long enough to be considered infinite, meaning that the zeromean hypothesis is correct. Second, we have assumed that the properties of turbulence are stationary, although they may vary with time in practice. The above derivation is valid only during the period over which this assumption is correct. These expressions can be upgraded for nonstationary turbulence by summing the lobes over finite timescales where turbulence can be assumed to be stationary. The underlying differences between the photometric and interferometric lobes will therefore remain. When the baseline between the two apertures is larger than the outer scale of the turbulence, as is the case for GRAVITY on the Unit Telescopes, the phase in the two apertures becomes uncorrelated because the structure function D_{ϕ}(u) saturates to . Contrary to the singlemode photometric lobe, the angular size of the singlemode interferometric lobe does not change and only becomes attenuated by the coherent energy. This behaviour is similar to the non singlemode case, as observed by Rousset et al. (1992), and was first demonstrated by Mège (2002).
As shown by Guyon (2002) in the case of a point source, the visibility function of the apodised object can be recovered if the coherent energy can be accurately calibrated. The loss of the signaltonoise ratio on the visibility measurements increases exponentially with turbulence strength, however. Guyon (2002) also showed that the traditional calibration becomes difficult to impossible when the object size is comparable to or larger than the fibre mode. An unresolved calibrator will not allow calibrating the widening of the photometric lobe. It is then mandatory to measure the lobe in all the directions of the object with the same exposure time, assuming turbulence characteristics have not changed in the meantime. This is discussed in Sect. 5.4.
5. Effects of partial phase correction
5.1. Effect on the interferometric field of view
Singlemode interferometers are rarely used in fully turbulent conditions. The phase variations across the telescope apertures are either partially corrected for by an adaptive optics system and/or modified by additional perturbations of instrumental origin. As an illustration, we consider a scenario where the atmospheric turbulence is almost fully corrected for by an adaptive optics system, and the phase disturbance is dominated by turbulence, mostly tiptilt, generated inside the optical train of the interferometer. This is the case when a bright object is observed with Very Large Telescope Interferometer (VLTI)/GRAVITY (Gravity Collaboration 2017). Although the assumption that tunnel turbulence shares the behaviour of free air turbulence is quite strong, it is reasonable to assume that the statistics of tiptilt residuals are stationary and average down to zero over relatively short timescales.
The phase in the pupil of telescope i writes
where t_{i} = (t_{i, x}, t_{i, y}) are the timevarying angular tip and tilt coefficients and p_{i} is the timevarying piston term. The instantaneous cross correlation of the apodised pupils function then writes
The tip, tilt, and piston are assumed to be independent and to follow Gaussian distributions. The standard deviations are σ_{t} for each tiptilt axis and σ_{p} for the differential piston. If the turbulence is decorrelated between the two pupils, the longexposure average ⟨⟩_{long} of the above expression writes
where u denotes the norm of u.
The average tiptilt effect is to multiply the apodised pupils by Gaussian functions of the spatial frequencies. For a perfectly Gaussian fibre mode in image space,
where k is adjusted^{1} to maximise the coupling of astronomical light from an unresolved object to a singlemode fibre in the diffractionlimited case (Coudé du Foresto et al. 2000), the longexposure apodised pupil affected by tiptilt writes:
The effect of the residual tiptilt is to further narrow the width of the apodised pupil function, whose Fourier transform is consequently enlarged. A new interferometer lobe function can be defined in the same way as the fibre lobe was defined in Eq. (16):
In absence of tiptilt residues, the interferometer lobe matches the aberrationfree fibre lobe. In the general case, however, its onsky full width at half maximum (FWHM) increases with the tiptilt amplitude as follows:
This result is new and has fundamental implications for the observation with singlemode interferometers of extended objects such as the Galactic centre. Sources outside the diffraction limit of a single telescope but inside FWHM_{t} do contribute to the coherent flux. Illustrations of this effect on the Unit Telescopes of the VLTI for different levels of residual tiptilt are presented in Fig. 2. We recall that the uncorrected tunnel tiptilt at the VLTI is of the order of ∼40 mas per axis in the worst case, but is reduced to about ∼10 mas per axis when the tiptilt is controlled with the acquisition camera of GRAVITY.
Fig. 2. Top: normalised profile in pupil plane of a Gaussian fibre mode (blue) that optimally matches a VLTI/UTlike 8 m telescope pupil (black) where the central obscuration has been neglected. The additional tiptiltinduced apodisation profiles are given for 10, 20, and 40 mas. The apodisation function is the product of the fibre mode and of the tiptilt apodisation function. Middle: interferometric lobes for 0 mas, 10, 20, and 40 mas of tiptilt, shown in logarithmic scale. Each lobe amplitude is normalised on axis and therefore does not illustrate the coupling performance reduction as the tiptilt perturbation increases. As shown in the text, the photometric lobe matches the interferometric lobe when tiptilt dominates. Bottom: ratio between the 10, 20, and 40 mas tiptilt lobes and the lobe without tiptilt, shown in logarithmic scale. All tiptilt rms values are for the total tiptilt in all figures. 
The phase variance of the modes of turbulent phase decreases with increasing order so that in practice, residual tiptilt contains most of the phase variance after correction. The influence of higher modes is therefore expected to be low at high Strehl ratio, as is the case with the Coudé Infrared Adaptive Optics (CIAO) adaptive optics system of GRAVITY. To illustrate this, we now consider, in addition, the residual turbulent phase measured by the infrared wavefront sensor CIAO when the loop on IRS7 is closed in the Galactic centre. In the particular case of CIAO and GRAVITY, Fig. 3 shows that the effect of modes higher than tiptilt is rather negligible on the apodisation of the pupil and therefore on the interferometric field of view.
Fig. 3. Pupil apodisation associated with the CIAO residual turbulence for a typical observation of the Galactic centre. There is an indication that the low orders (tiptilt) are less strongly corrected for that the higher orders. Overall, the residual turbulence is not responsible for any significant widening of the interferometric field of view. 
The tiptilt effect can be intuitively explained. The highorder turbulence modes do not affect the image position to zeroth order but destroy the spatial coherence of light, with the consequence that speckles of various phases are introduced in the image. The tiptilt has a different effect. It does not affect the instantaneous spatial coherence and only shifts the object position in the image plane. In the twotelescope case, when the two images overlap, or equivalently, when the differential tiptilt is of the order of the diffraction limit at most, then spatial coherence is high. For a binary object where one of the components is at the centre of the turbulencefree field of view, the two objects are equally affected by tiptilt, but the effect on spatial coherence is different. At the centre of the field of view, the lobe transmission is flat and maximum, and therefore the tiptilt effect is that spatial coherence is reduced. At the edge of the lobe, the throughput variation is steeper, and any positive tiptilt fluctuation (i.e. in the field centre direction simultaneously in both arms of the interferometer to ensure a good overlap of the images) leads to a strong increase in coherence throughput.
5.2. Tiptilt effect on the field of view around Sgr A*
The net effect of widening the field of view by tiptilt from tunnel seeing on the photometric ratio of Sgr A* and S2 in the case of GRAVITY is represented in Fig. 4. We have used K magnitudes of 14.2 for S2 and 16.5 for Sgr A* in a quiescent state (Genzel et al. 2003), whereas the K magnitude reached 15.5 for the first moderate flare detected by GRAVITY (Gravity Collaboration 2017) and can be as high as 14.5 for a brighter flare (Genzel et al. 2003). The angular distance of S2 was set to various values ranging from 11 mas, the closest distance at pericentre is 15 mas according to Gillessen et al. (2009), up to 84 mas at the distance at the time of the observation reported in Gravity Collaboration (2017).
Fig. 4. Fieldofview effect: widening by tiptilt residues on the contrast ratio between S2 and a quiescent Sgr A* as a function of relative distance and tiptilt residue rms. 
The Sgr A*toS2 flux ratio is plotted for various tiptilt rms values and as a function of fibre offset from Sgr A* in the direction of S2. The rms tiptilt measured by GRAVITY is typically 15 mas under good seeing conditions, meaning that it is limited by tunnel seeing. The plots show the net tiptilt effect: it increases the relative contributions of the sources in the field of view around the central source. For a separation of 84 mas, the Sgr A*toS2 flux ratio is roughly 10^{4} in absence of tiptilt and with the fibre centred on Sgr A*. This same flux ratio drops to 10^{2} for a 15 mas tiptilt and reaches 1 when a fibre offset of 20 mas is additionally applied. Four orders of magnitude can therefore be compensated for with moderate tiptilt and fibre offset. For shorter separations, the Sgr A*toS2 flux ratio is either systematically of the order of 1 or lower, with moderate rms tiptilt applied.
Although the widening of the field of view found in Sect. 5 is small, the effect is quite large when the source at the centre of the field is much fainter than the source at the edge. This clearly explains why S2 was detected and appeared so bright in the image reconstructed in Gravity Collaboration (2017), despite the 85 mas offset.
5.3. Strategies to widen the field of view
Another way to look at the tiptilt effect on an interferometric field of view in the case of the Galactic centre is to study the apparent magnitude of S2, that is, the magnitude of S2 attenuated by the interferometric lobe function as it appears in a reconstructed image. The current sensitivity of the reconstructed images of GRAVITY is K = 17.5 (Gravity Collaboration 2017), and it will probably increase in the future. Figure 5 gives the apparent magnitude of S2 as a function of rms tiptilt and distance to Sgr A*, assuming the fibre is centred on Sgr A*. Setting a threshold of K = 18 for S2, we can read the rms tiptilt required to detect S2 from this figure. It increases with the separation between S2 and Sgr A*. 50 mas of rms tiptilt is required to detect S2 for a distance of 100 mas. This indeed provides a method to widen the field of view of the interferometer: apply random tiptilt during exposures. However, the effect of bandwidth smearing needs to be taken into account in this case because Sgr A* and S2 are observed in the most sensitive mode of GRAVITY with the lowest spectral resolution R = 22 with an intrinsic FWHM of the interferometric field of view of order 60 mas (). The widening of the field of view through tiptilt also affects the central source, as shown in Fig. 6. The central source is all the dimmer as the tiptilt rms is higher. When in a quiescent state, Sgr A* becomes as low as the threshold of K = 18 for an rms tiptilt of 50 mas. All in all, given the current sensitivity of the reconstructed images of GRAVITY, the field of view can be extended up to 120 mas with an artificial tiptilt of 37 mas. In addition to tiptilt, the apparent magnitude of S2 in the reconstructed images can be increased by moving the fibre towards S2. Figure 7 shows the fibre offset that is to be applied to reach certain threshold magnitudes for S2 as a function of rms tiptilt. The offsets were computed for four different threshold magnitudes: 16, 17, 18, and 19. The higher
Fig. 5. Tiptilt effect on the apparent magnitude of S2 in the images reconstructed by GRAVITY centred on the Galactic centre. A 50 mas tiptilt allows keeping the apparent magnitude of S2 above the current sensitivity threshold of K = 18 in the images reconstructed from GRAVITY data, up to a separation of ∼100 mas. 
Fig. 6. Tiptilt effect on the attenuation of the interferometric throughput at the centre of the field of view. The figure also shows the level of attenuation where a quiescent K = 16.5 or bright flaring K = 14.5 Sgr A* reaches the assumed K = 18 detection limit. A quiescent Sgr A* would no longer be observable above ∼37 mas tiptilt rms, whereas a bright flare would still be observable past 60 mas. 
Fig. 7. Fibre offset needed to reach the threshold magnitude to detect S2 in the reconstructed GRAVITY images as a function of rms tiptilt. Results are plotted for four different threshold magnitudes: 16, 17, 18, and 19. 
the threshold (and therefore the sensitivity of the GRAVITY reconstructed maps), the lower the required offset. For a threshold of K = 18, no fibre offset is required up to a separation between S2 and Sgr A* of 70 mas. Above this separation, the offset required to detect S2 increases with decreasing rms tiptilt. Another similar method is to perform a 2D scan of the field of view in a single exposure. This will be discussed in a forthcoming paper. This is an alternative to the wellknown mosaicing method of radio astronomy.
5.4. Effect on photometric calibration
A similar computation as in the previous section is applied to the photometric lobe when turbulence is dominated by tiptilt alone. In this case, the structure function of the turbulent phase takes the simple expression
with the longexposure photometric mode of Eq. (20) becoming
As a consequence, the longexposure photometric mode does not match the longexposure interferometric mode of Eq. (26). This is confirmed by the plots in Fig. 8. The reason for this is the noncorrelation of the tiptilt between the two interferometric pupils. The effect is less dramatic than in Eq. (20) for the case of uncorrected turbulence, however, where the interferometric lobe shape is independent of turbulent while the photometric lobe increasingly widens with increasing turbulence strength. When the FWHM is computed for the 10 mas rms tiptilt case, the difference is not so great, however (70 mas versus 67 mas; 65 mas in both cases without turbulence).
Fig. 8. Photometric lobes for 0 mas, 20 mas, and 40 mas of tiptilt, shown in logarithmic scale. Each lobe amplitude is normalised on axis and therefore does not illustrate the coupling performance reduction as the tiptilt perturbation increases. The interferometric lobes of Fig. 2 are plotted as dashed lines. 
As pointed out by Guyon (2002), this difference between the interferometric and photometric lobes can make the photometric calibration of the singlemode visibilities difficult. In the case of GRAVITY, with the low rms of residual tiptilt and if the pointing of the fibre is accurate enough, the difference in lobe FWHM is so low that the effect could be negligible. However, if the contrast between the central source and the sources at the edge of the field of view is high (as is the case if Sgr A* is at the centre of the field and in quiescent or lowstate mode), then the photometric correction of the visibilities can lead to a substantial visibility bias. More photons are coupled to the photometric lobes than to the interferometric lobes, meaning that the number of coherent photons is overestimated.
This can be shown by again expanding the twotelescope balanced interferogram of Eq. (12), but keeping the photometric terms:
Injecting the photometric and interferometric lobes of Eq. (10) and Eq. (17), respectively, results in
Applying the Zernike–Van Cittert to the third term, we obtain
where V_{12} is the visibility of the object that is apodised by the interferometric lobe. When we assume that the photometric lobes are identical and equal to L in the two beams, then the measured visibility writes
and the visibility multiplicative bias is therefore the coherenttouncoherent photometry ratio. As a consequence, the visibilities are underestimated, leading to values lower than 1 even at very short baselines. This effect is shown in Fig. 9, where it has been studied for different separations between Sgr A* and S2, with the former in different states of brightness (bright flare, moderate flare, and quiescent state). We have assumed that the fibres are perfectly centred on Sgr A* in the two beams of the interferometer. The effect is almost negligible for a separation of 11.1 mas up to a residual tiptilt amplitude of 20 mas because the contrast loss is 1.1% in the quiescent state. In all other cases, however, for larger separations, the contrast loss is much higher than the accuracy of the instrument. It is close to 20% at 20 mas residual tiptilt for a separation of 59.8 mas in the quiescent state. A correction has to be applied to visibilities if the rms tiptilt can be estimated or the bias has to be modelled to fit the data. In either case, it is necessary to assess the effect of residual tiptilt on lobes either through a calibration of the lobes or by measuring the tiptilt variance. The nonstationarity of seeing could be a problem here, which would need to be addressed in a separate study.
Fig. 9. Effect of differential interferometric vs. photometric fieldofview widening caused by residual tiptilt. The figure shows the maximum visibility measured for the S2/Sgr A* pair as a function of tiptilt perturbation for four different separations between the two sources and for the three different states of Sgr A* (top: bright flare; middle: moderate flare; bottom: quiescent state). 
6. Conclusion
The field of view of an interferometer is defined as the field in which sources contribute to spatial coherence or interferences. It is constrained by various parameters that range from hardware limitations to spectral resolutions and (u, v) plane sampling. Here, we focused on the special case of singlemode interferometers where the maximum field of view is set by the fibre mode. We explored for the first time the influence of partial correction in adaptive optics on the maximum field of view of singlemode interferometers at optical wavelengths in the longexposure regime. This is a new regime opened by the GRAVITY instrument of the VLTI. Past studies were made by Guyon (2002) in the shortexposure regime with an application to binaries and by Mège (2002), but were restricted to the case without phase correction. Like these authors, we defined photometric and interferometric lobes, as is also done for radio interferometers. We also found that the two lobes are not identical, except in absence of turbulence or when the distance between the pupils of the interferometer is much larger than the outer scale of turbulence. In the turbulent case without correction, the spatial distribution of the interferometric lobe is identical to the aberrationfree case. In these two cases, the maximum field of view of the interferometer matches the diffraction limit of a single aperture. The situation is different when turbulent aberrations are partially corrected for. We have studied the particular case where the wavefront residues are dominated by tiptilt. We have shown that the field of view is widened by tiptilt residues, with a specific application to the Galactic centre, where stars at more than one diffraction limit away from the centre can contribute to the interferometric signal. Although the widening of the field of view remains modest for tiptilt residues of 10 mas per axis, it dramatically changes the measured visibility when the central object is much fainter than the surrounding sources. In practice, the field of view can be more than doubled. We have also shown that the field of view can be extended by generating dynamical tiptilt aberrations. Because the photometric and interferometric lobes do not match under turbulent conditions, we have discussed the effect of this issue on the calibration of visibilities when tiptilt residuals dominate and have shown that the effect can be calibrated if the rms residual tiptilt is known.
Last, we focused this paper on the amplitude of the photometric and interferometric lobes. However, the phase of the singlemode interferometric lobes also plays an important role by potentially affecting the phase of the interferometric observables. Intuitively, the more completely the observed object fills the interferometric lobe, the more its imaging and/or astrometry is affected. We leave this investigation to a forthcoming paper.
For an optimal coupling between a singlemode Gaussian profile and a telescope, the size of the Gaussian beam k has to match the telescope diameter D following , where λ is the wavelength and ϵ a factor that depends on the shape of the pupil or on the central obscuration if the pupil is circular. The ϵ factor is close to unity, and the FWHM of the fibre mode almost perfectly matches the diffraction limit.
Acknowledgments
The authors are grateful to the referee for the very useful interaction that helped us to improve the paper. This work was supported by Paris Observatory, by the Programme National Cosmologie et Galaxies (PNCG) of CNRS/INSU with INP and IN2P3, cofunded by CEA and CNES and by the Programme National GRAM of CNRS/INSU with INP and IN2P3, cofunded by CNES.
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All Tables
Summary of the flux and coherence quantities measured with singlemode telescope arrays.
All Figures
Fig. 1. Telescope lobe measured on an internal source (left) and on a star with partial adaptive optics correction (right), obtained with the ‘OHANA experiment (Perrin et al. 2006) on one of the telescopes of the Keck Interferometer (Colavita et al. 2013). 

In the text 
Fig. 2. Top: normalised profile in pupil plane of a Gaussian fibre mode (blue) that optimally matches a VLTI/UTlike 8 m telescope pupil (black) where the central obscuration has been neglected. The additional tiptiltinduced apodisation profiles are given for 10, 20, and 40 mas. The apodisation function is the product of the fibre mode and of the tiptilt apodisation function. Middle: interferometric lobes for 0 mas, 10, 20, and 40 mas of tiptilt, shown in logarithmic scale. Each lobe amplitude is normalised on axis and therefore does not illustrate the coupling performance reduction as the tiptilt perturbation increases. As shown in the text, the photometric lobe matches the interferometric lobe when tiptilt dominates. Bottom: ratio between the 10, 20, and 40 mas tiptilt lobes and the lobe without tiptilt, shown in logarithmic scale. All tiptilt rms values are for the total tiptilt in all figures. 

In the text 
Fig. 3. Pupil apodisation associated with the CIAO residual turbulence for a typical observation of the Galactic centre. There is an indication that the low orders (tiptilt) are less strongly corrected for that the higher orders. Overall, the residual turbulence is not responsible for any significant widening of the interferometric field of view. 

In the text 
Fig. 4. Fieldofview effect: widening by tiptilt residues on the contrast ratio between S2 and a quiescent Sgr A* as a function of relative distance and tiptilt residue rms. 

In the text 
Fig. 5. Tiptilt effect on the apparent magnitude of S2 in the images reconstructed by GRAVITY centred on the Galactic centre. A 50 mas tiptilt allows keeping the apparent magnitude of S2 above the current sensitivity threshold of K = 18 in the images reconstructed from GRAVITY data, up to a separation of ∼100 mas. 

In the text 
Fig. 6. Tiptilt effect on the attenuation of the interferometric throughput at the centre of the field of view. The figure also shows the level of attenuation where a quiescent K = 16.5 or bright flaring K = 14.5 Sgr A* reaches the assumed K = 18 detection limit. A quiescent Sgr A* would no longer be observable above ∼37 mas tiptilt rms, whereas a bright flare would still be observable past 60 mas. 

In the text 
Fig. 7. Fibre offset needed to reach the threshold magnitude to detect S2 in the reconstructed GRAVITY images as a function of rms tiptilt. Results are plotted for four different threshold magnitudes: 16, 17, 18, and 19. 

In the text 
Fig. 8. Photometric lobes for 0 mas, 20 mas, and 40 mas of tiptilt, shown in logarithmic scale. Each lobe amplitude is normalised on axis and therefore does not illustrate the coupling performance reduction as the tiptilt perturbation increases. The interferometric lobes of Fig. 2 are plotted as dashed lines. 

In the text 
Fig. 9. Effect of differential interferometric vs. photometric fieldofview widening caused by residual tiptilt. The figure shows the maximum visibility measured for the S2/Sgr A* pair as a function of tiptilt perturbation for four different separations between the two sources and for the three different states of Sgr A* (top: bright flare; middle: moderate flare; bottom: quiescent state). 

In the text 
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