© ESO, 2010

1. Introduction

Performance of groundbased optical interferometers is severely limited by the atmospheric turbulent piston, which introduces a random optical path difference between the beams that are combined to produce fringes. These fringes are thus randomly moving on the detector, blurring the signal and preventing us from integrating on time longer than the coherence time of the atmosphere, typically a few tens of milliseconds in the infrared. As a consequence, the limiting-magnitude and ultimate precision of groundbased interferometers are dramatically reduced.

By estimating and compensating in real-time the interferometric phase – in other words by “locking” the fringes on the detector, fringe tracking and phase referencing devices are powerful instruments to circumvent this problem, allowing us to integrate the interferograms on much longer time frames. These instruments, which noticeably improve the sensitivity and the accuracy of interferometers, are undergoing major developments in recent few years, as the number of concepts that are currently studied for various interferometers (Le Bouquin et al. 2008; Sahlmann et al. 2008: VLTI; Berger et al. 2008: CHARA; Jurgenson et al. 2008: MROI) can attest.

In all the different concepts proposed, one key issue still in debate among the instrumental community is the relevance of using spatial filtering devices such as singlemode fibers or integrated optics to carry/combine the beams. It is often argued that spatial filtering is not suited for that type of instruments,

because only a fraction of the total flux is injected in the spatial filter component. This coupling efficiency, which is typically on the order of the Strehl ratio (Coudé du Foresto et al. 2000), can indeed be low with strong turbulence and/or poor adaptive optics (AO) correction. However, the case is not that simple. Singlemode filters only keep that part of the incoming flux which is related to the coherent part of the corrugated wavefront. In other words, singlemode devices only propagate the first mode of the electro-magnetic field, leaving at its output a plane wavefront – that is a deterministic signal, the price to pay is a loss of flux that is correlated to the strength of the turbulence. At the contrary, multimode schemes such as bulk optics are preserving the total flux, but keep the incoherent part of the wavefront at the output (producing randomly moving speckle patterns in the image), which makes the interferogram sensitive to turbulence. Hence choosing between singlemode and multimode schemes corresponds to decide whether losing flux or losing coherence in the signal will be the best strategy to optimize the performance of the interferometer.

Indeed, the efficiency of singlemode devices in terms of precision and robustness of the estimation of the amplitude of the visibility has already been demonstrated experimentally (Coudé du Foresto et al. 1997) and theoretically as well (Tatulli et al. 2004). But concerning the estimate of the phase – which is the quantity of interest for fringe tracking and phase referencing instruments, the situation is not clear yet. If simulations have been initiated in some specific cases to estimate the effects of spatial filtering (Buscher et al. 2008: phase jumps;

Tubbs 2005: computation of the coherence time), no theoretical formalism regarding this problem has been presented so far to our knowledge.

We here derive the error of the interferometric phase with detector, photon, and atmospheric noises, both for the singlemode and multimode cases in presence of partial AO correction, following a formalism analogous to the one previously developed for the squared visibility in Tatulli et al. (2004). We then compare the performance of singlemode and multimode schemes for the estimate of the phase of the interferograms, and apply our analysis in the framework of fringe-tracking and phase referencing methods.

2. The phase in groundbased interferometry

2.1. General description and underlying assumptions

The groundbased interferometer: As sketched in Fig. 1, we consider an interferometer with two telescopes, separated by the baseline distance B₁₂. We assume that both telescopes are identical, that is they are described by the same pupil function P(r) of diameter D, therefore by the same collecting area Σ_P defined as Σ_P = ^∫ [ P(r) ] ²dr. Their transmissions are different however, namely t₁ and t₂ respectively. The wavefront over each telescope is affected by randomly moving phase screens, φ₁(r) and φ₂(r) respectively. The typical cell size of these turbulent phases is r₀ (Fried 1965) and the strength of the turbulence is characterized by the quantity D/r₀. When Adaptive Optic systems are installed on each telescope, they partially compensate in real-time the wavefront corrugations of the atmosphere. The correction is not perfect however, and residual phases and remain, affecting the two telescopes. Because of this turbulence, the optical path difference (opd) between the two light paths to the star through the two telescopes is randomly shifted by the so-called turbulent piston , i.e. the phase difference between the two telescopes averaged over the pupils: ${\mathit{\phi }}_{\mathrm{12}}^{\mathit{p}}\mathrm{=}\mathrm{\int }\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{-}\mathrm{\int }\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathit{.}$(1)The interferogram: A 2-telescope interferogram consists in an incoherent part, which is the sum of the photometric fluxes coming from both telescopes, and a modulated coherent part – namely the fringes, which are proportional to the so-called complex coherent flux , which depends on the flux N (photons per surface unit and per time unit) and on the complex visibility V₁₂ of the source corresponding to the baseline frequency f₁₂ = B₁₂/λ of the interferometer. λ is the effective wavelength of the interferogram. The modulation of the fringes can be either temporal, when the opd is scanned with moving piezo-mirrors (temporal coding), or spatial, when the opd is scanned with dedicated output pupils, whose separation defines the frequency of the modulation (spatial coding).

Finally, the phase of the interferogram Φ₁₂, that is the phase shift with respect to the zero opd, originates from the source phase , a potential instrumental phase , which we will assume equal to zero below¹, and the turbulence piston phase  .

2.2. Estimating the interferometric phase and its associated error

Estimating the interferometric phase Φ₁₂ requires us to define and compute from each interferogram an appropriate complex estimator of the complex coherent flux . Several methods are possible to build this estimator, such as “ABCD” techniques in the image plane (Colavita 1999; Tatulli et al. 2007) or an analysis in the Fourier plane (Roddier & Lena 1984; Coudé du Foresto et al. 1997). In any case, the measured phase is then the argument of the complex estimator : ${\mathrm{\Phi }}_{\mathrm{12}}\mathrm{=}\mathrm{atan}\left[\frac{\mathrm{Im}\mathrm{(}{\begin{array}{}\mathrm{􏽧}\\ {\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\end{array}}^{\mathrm{)}}}{\mathrm{Re}\mathrm{(}{\begin{array}{}\mathrm{􏽧}\\ {\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\end{array}}^{\mathrm{)}}}\right]\mathrm{·}$(2)In this framework, one can show that in first approximation, i.e. when the error on is small compared to its amplitude, the variance of the instantaneous phase (i.e. measured over one interferogram) can be expressed as (Chelli 1989, see Eq. (5), case j = k): ${\mathit{\sigma }}_{\mathit{\phi }}^{\mathrm{2}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathit{E}\left({\left|\begin{array}{}\mathrm{􏽧}\\ {\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\end{array}\right|}^{\mathrm{2}}\right)\mathrm{-}\mathrm{Re}\left[\mathit{E}\left({\begin{array}{}\mathrm{􏽧}\\ {\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\end{array}}^{\mathrm{2}}\right)\right]}{{\mathrm{[}\mathit{E}{\mathrm{(}{\begin{array}{}\mathrm{􏽧}\\ {\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\end{array}}^{\mathrm{)}}}^{\mathrm{]}}}^{\mathrm{2}}}\mathit{,}$(3)where E denotes the expected value. Chelli (1989) has also demonstrated that the variance of the interferometric phase is independent of the object phase. For the sake of simplicity, we will therefore consider that the source of interest is centro-symmetric, namely .

Regardless of the method used to build the estimator from the interferogram, the statistics of the phase will depends on whether the interferometer is using a multimode or a singlemode design. Below, we explore how the atmospheric spatial fluctuations of the turbulent wavefront affect the fringe pattern in multimode and singlemode cases.

2.3. The coherent flux in multimode interferometry

In multimode interferometry, the total flux on the detector remains constant (neglecting scintillation) and the interferograms are created in speckle patterns randomly moving with the fluctuations of the turbulent wavefronts over the two telescopes.

There are two different ways to combine the beams in multimode interferometry (Chelli & Mariotti 1986): whether in the image plane, a technique which is suited to perform spatial coding of the fringes on the detector, known as Fizeau (Beckers & Hege 1984) and Michelson (Mourard et al. 1994) mountings², or in the pupil plane, which is commonly used to temporally sample the interferogram through dedicated moving piezo-electric mirrors (Dyck et al. 1995). In both cases however, the expression of the coherent flux can be written as (as shown in the demonstration of Appendices A.3.1 and A.3.2 of this paper and in Appendices A1 and A2 of Buscher et al. 2008): ${\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}{\mathit{T}}_{\mathrm{12}}\mathit{.}$(4)T₁₂ is the normalized interferometric transfer function resulting of the cross-correlation of the pupil of the two telescopes, corrugated by the residual atmospheric screen phases and , (see Fig. 1). It writes: ${\mathit{T}}_{\mathrm{12}}\mathrm{=}\frac{\mathrm{\int }\mathrm{\left[}\mathit{P}\mathrm{\right(}{r}\mathrm{)}{\mathrm{]}}^{\mathrm{2}}{\mathrm{e}}^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}}{\mathrm{\int }\mathrm{\left[}\mathit{P}\mathrm{\right(}{r}\mathrm{)}{\mathrm{]}}^{\mathrm{2}}\mathrm{d}{r}}\mathrm{·}$(5)As an analogy with classical AO systems on single-pupil telescopes (see e.g. Fusco & Conan 2004),  |T₁₂| ² can be seen as the instantaneous interferometric Strehl ratio, for multimode interferometry.

2.4. The coherent flux in singlemode interferometry

The main property of singlemode devices such as fibers or integrated optic chips is to perform a spatial filtering of the input wavefront so that only its Gaussian part is transmitted. As a consequence, a singlemode device turns the input spatial wavefront fluctuations into intensity fluctuations at the output. With this, each outgoing wavefront is flat, hence the shape of the interferogram is deterministic, only depending on the instrumental configuration. The trade-off is that only a fraction of the flux is transmitted, corresponding to the coherent energy of the turbulent wavefront. The singlemode instantaneous coherent flux thus takes the form (see Appendix A.4, Eq. (A.38)) ${\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}^{\mathrm{12}}\mathrm{\left(}\mathit{V}\mathrm{\right)}\mathit{,}$(6)where ρ¹²(V) is the interferometric coupling coefficient that depends both on the source extension and on the level of AO correction (Tatulli et al. 2004, see Eqs. (3), and (A.35), Appendix A.4 of this paper). Focusing on compact sources, that is astrophysical objects unresolved by a single telescope, ρ¹²(V) takes a simple expression of the form ρ¹²(V) = ρ₀V₁₂ρ₁₂ (Tatulli & Chelli 2005, Eqs. (4–6)). The coherent flux thus rewrites: ${\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}_{\mathrm{0}}{\mathit{V}}_{\mathrm{12}}{\mathit{\rho }}_{\mathrm{12}}\mathit{,}$(7)where ρ₁₂ is independent of the source properties ${\mathit{\rho }}_{\mathrm{12}}\mathrm{=}\frac{\mathrm{\int }\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{e}}^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}}{\mathrm{\int }\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\frac{\mathrm{\int }\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{e}}^{\mathrm{-}\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}}{\mathrm{\int }\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathit{,}$(8)and where ρ₀ is the maximum achievable coupling efficiency, shown to be  ~80% (Shaklan & Roddier 1988), because of the geometrical mismatch between the telescope Airy disk profile and the Gaussian profile of the propagated mode.

Equation (8) has to be compared with Eq. (5). We can see that they are almost similar, but with one major difference: in Eq. (5), the product of the phasors is integrated over the pupil, whereas in Eq. (8), each phasor is first integrated over the pupil, then the product is performed. As noticed by Buscher et al. (2008), this difference is the mathematical expression of the property of spatial filtering.

Again,  |ρ₁₂| ² – which is equal to 1 for perfect AO correction/absence of atmospheric turbulence – can be seen by as the instantaneous interferometric Strehl ratio, but for the singlemode case.

3. Performance comparison

3.1. Phase noise

From Eqs. (3–7), we derive the expression of the error of the phase for both singlemode and multimode cases, as developed in Appendix A. We show that the variance of the interferometric phase can be decomposed as the quadratic sum of three terms, corresponding to the detector, photon, and atmospheric regimes: ${\mathit{\sigma }}_{\mathit{\phi }}^{\mathrm{2}}\mathrm{=}{\mathit{\sigma }}_{{\det }_{\mathit{\phi }}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{{\mathrm{phot}}_{\mathit{\phi }}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathrm{at}{\mathrm{m}}_{\mathit{\phi }}}^{\mathrm{2}}\mathit{,}$(9)where the detail of each term is given in Table 1. The phase error depends on parameters coming from the source and the instrument, which we recall here: N is the number of incoming photons per surface unit and per time unit,  |V₁₂|  is the amplitude of the visibility, t₁ and t₂ are the transmissions of both telescopes, N_pix is the number of pixels to sample the interferogram, σ_det is the detector noise. The phase error also depends on atmospheric terms: the long exposure Strehl ratio and its associated coherent energy where is the variance of the residual phase of a single telescope (Noll 1976), and of the phase structure function with partial AO correction as modelled in Appendix D.

Figure 2 (top) shows the multimode and singlemode phase error as a function of the number of incoming photoevents (alternatively, source K-band magnitude), for different levels of AO correction (i.e. different Strehl ratios). In a general way, the behavior of the error is similar in both cases: at low fluxes, the detector noise dominates with a slope of the error in 1/N, then it goes through a (short) photon noise regime in and eventually reaches a plateau for the brightest sources due to speckle noise – analog to the one known in the case of the visibility (Goodman 1985; Tatulli et al. 2004), which limits the ultimate precision on the phase. Note that in order to keep the phase noise below reasonable levels, that is smaller than 1 radian, it is mandatory to equip interferometers with AO systems, to ensure at least a low-order correction of the turbulent phase, providing Strehl ratios greater than  ~0.1 for D/r₀ = 8 and  ~0.5 for D/r₀ = 2.

Going into greater detail, Fig. 2 (bottom) shows the ratio of the error of the multimode phase by the error of the singlemode phase as a function of the number of photoevents. Clearly, two different behaviors occur if we are considering a photon-starved or a photon rich regime, as discussed below.

3.2. Low-light level regime – limiting magnitude

For faint sources, the phase error is dominated by the detector noise σ_detφ. The comparison between both multimode and singlemode cases is not straightforward however because the number of pixels required to sample the fringes is specific to each technique, and depends on the chosen combination scheme. Let us first review here the possible technical solutions:

Multimode – (image plane) multiaxial combination: Here, each speckle in which the interferograms are formed must be sampled correctly, that is it must be crossed by at least two pixels (Chelli & Mariotti 1986). Then, if we take the whole image of size λ/r₀, the total number of pixels required becomes N_pix = 2D/r₀. As a consequence, N_pix is dependent on the turbulence, in this combination mode. In the case of the VLTI at Cerro Paranal, the average r₀ is  ≃1m in the K-band, which gives turbulence strengths of D/r₀ ≃ 8 and D/r₀ ≃ 1.8 for the UTs (D = 8 m) and ATs (D = 1.8 m). This corresponds to a number of pixels that can vary quite a lot with the strength of the turbulence, N_pix = 16 (UTs) and N_pix = 4 (ATs) respectively.

Multimode – (pupil plane) coaxial combination: Providing that the interferogram is scanned faster than the coherent time of the atmosphere in order to “freeze” the fringes, the minimum number of pixels to obtain full information on the temporal interferogram is 3, which corresponds to the 3 degrees of freedom: incoherent flux, fringe amplitude, and phase. However, instead of this “ABC” scheme with 120 degrees between the three channels, it is frequently more practical to implement an “ABCD” scheme with 90 degree phase shifts, hence requiring 4 pixels. Note that in the framework of phase tracking, it is also possible to only measure the sine component of the fringe with an “AC” scheme with 180 degrees between the two channels. In any case, the number of pixels needed in a coaxial pupil plane combiner is between 2 and 4.

Singlemode – multiaxial combination: Contrary to the multimode/multiaxial combination, the number of pixels here is independent of the turbulence, because the shape and frequency of the interferogram are fixed by the design of the beam combiner. Typically, the interferogram consists in a sinusoidal signal with a frequency defined by the separation of the output pupils (the so-called coding frequency), and where its amplitude is modulated by the Gaussian envelope of the singlemode device. Tatulli & LeBouquin (2006) have shown that the optimum number of pixels that respects the Shannon criterion (>2 pixels per fringe) and prevents an overlapping of the photometric and interferometric peaks in the Fourier plane is N_pix = 10.

Singlemode – coaxial combination: As in the coaxial multimode case, one merely needs to sample the interferogram with respect to the 3 degrees of liberty here. As a result, the number of pixels is again between 2 and 4. Note that instead of the usual temporal coding, fringes can be scanned simultaneously thanks to “ABCD-like” integrated chip devices (Benisty et al. 2009).

As a consequence, coaxial schemes – both in multimode and singlemode cases – appear more appropriate, because they are using substantially fewer pixels than multiaxial ones. We remark however that these conclusions apply in the framework of fringe tracking where pair-wise combinations are favored. They may differ in the context of an interferometric imaging instrument where an important number of baselines is involved. Below we will consider coaxial combiners with N_pix = 4, corresponding to the standard “ABCD” sampling.

As multimode and singlemode fringe trackers finally require the same number of pixels, a straightforward comparison of the expressions of the detector noise shows that multimode combiners will in this regime achieve a slightly better performance by a factor of 1/ρ₀, because singlemode spatial filters cannot transmit 100% of the flux for the geometrical reasons mentioned in previous sections. If we define the limiting magnitude³ such as the error of the phase to be equal to 1 rad, which – apart from very low AO correction levels – occurs in this detector noise regime, the corresponding limiting flux is given by ${\mathit{K}}_{\lim }^{\mathrm{\left[}\mathrm{multi}\mathit{,}\mathrm{single}\mathrm{\right]}}\mathrm{=}\frac{\sqrt{\mathrm{2}}\sqrt{{\mathit{N}}_{\mathrm{pix}}}{\mathit{\sigma }}_{\det }}{\mathrm{\left[}\mathrm{1}\mathit{,}{\mathit{\rho }}_{\mathrm{0}}\mathrm{\right]}{\mathit{V}}_{\mathrm{12}}{\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathrm{r}}}^{\mathrm{2}}}}\mathit{,}$(10)with the factor 1 or ρ₀ depending whether we are using multimode or singlemode interferometers. As illustrated by Fig. 3, the gain in limiting magnitude for multimode combiners is  ~0.25 magnitude.

3.3. High-light level regime: the speckle noise

In the photon-rich regime, the speckle noise dominates and the error on the interferometric phase reaches a plateau, that is, is not dependent on the flux of the source anymore. As illustrated by Fig. 4, singlemode interferometers always provide a lower phase error than multimode ones, emphasizing the remarkable properties of spatial filtering of the turbulent wavefront by singlemode devices. This behavior was already noticed by Tatulli et al. (2004) for the estimate of the squared visibility, showing that the so-called modal speckle noise of the visibility was, for a given AO correction, always lower than classical speckle noise of multimode interferometers. The concept of modal speckle noise can also be applied for the singlemode interferometric phase. However, at the difference of the squared visibility for which the modal speckle noise is 0 for a point source, the phase modal speckle noise always exists, independently of the size of the source.

Note that the gain of using singlemode schemes is all the more important than the level of correction is low. If for a fairly good correction with a Strehl ratio above 20%, the difference remains marginal with a factor  ~1−1.5 between the singlemode and multimode phase error, the situations where bright sources are observed with low/none AO correction will highly benefit from singlemode interferometers. In these cases the precision of the phase can increase by at least a factor of 2 and much more when using spatial filtering of the corrugated wavefront with a typical Strehl ratio below 10%. This is a typical counter-intuitive example where it is more profitable to loose a substantial part of the flux and keep only the coherent part of the perturbed wavefronts than conserving the whole flux at the price of introducing additional atmospheric noise.

4. Application to fringe tracking

In this section, we apply the formalism developed above in the context of on-axis and off-axis fringe tracking, that is when the phase is estimated and compensated in real time to stabilize the fringes on the detector. We focus on the relative performance of fringe tracking systems using either multimode or singlemode schemes.

4.1. Coherent integration

For each interferogram, fringes are shifted from the zero optical path difference by the turbulent piston. This piston is corrected in real time by the fringe tracker, which estimates the phase of the fringes and compensates the optical path difference in real time by moving dedicated mirrors.

Centering the fringes in real time allows us to perform coherent integration of the signal, that is to integrate on time scales much longer than the coherence time of the atmosphere. However, the estimated piston used for the opd correction is affected by a random measurement error ϵ(t). As a result, the interferogram is not perfectly centered and is still moving with an excursion depending on the statistics of the noise. If the interferograms are integrated over a time that is long enough for the realizations of the random error ϵ(t) to span all the range of its probability law, then, by rules of ergodicity, the visibility will be affected by a loss of contrast⁴, which writes $\mathrm{⟨}\mathit{V}{\mathrm{⟩}}_{\mathit{t}}\mathrm{=}\mathrm{⟨}\mathit{V}{\mathrm{⟩}}_{\mathit{ϵ}}\mathrm{=}\mathit{V}{\mathrm{e}}^{\mathrm{-}\frac{{\mathit{\sigma }}_{\mathit{\phi }}^{\mathrm{2}}}{\mathrm{2}}}\mathit{,}$(11)where  ⟨  ⟩ _ϵ is the ensemble average over the realizations of the noise, and is the variance of this noise as computed in Sect. 3.1. Hence, for long integration times, an attenuation of the coherent flux (equivalently of the visibility) by a factor is introduced by the instantaneous turbulent phase, which is corrected only to the precision of its estimation. This bias therefore depends on whether singlemode or multimode fringe tracking is used.

For the high-light level regime, where the fringe tracking is mostly expected to work, Fig. 5 shows the attenuation of the visibility as a function of the Strehl ratio for both singlemode and multimode systems. We can see that acceptable loss of contrast, typically  ≳ 0.8, is achieved as soon as moderate AO corrections with Strehl  ≳ 0.1 (for D/r₀ = 8) are provided. On the contrary, for a lower performance of the AO system, the attenuation coefficient drops rapidly and the advantage of tracking the fringes is lost because of the uncompensated turbulent fluctuations of the phase over the pupils.

One important conclusion to draw is that for a given AO correction, the visibility attenuation induced by the noise of the phase is always smaller when using singlemode fringe tracking systems instead of multimode ones. In other words, the maximum achievable atmospheric contrast is always higher using singlemode fringe tracker than with multimode systems. This is especially true for low AO correction cases where the attenuation can become twice as large and more, as shown in Fig. 5.

4.2. Phase jumps

It is usually implicitly assumed that the phase estimated in interferometry is effectively the shift of the fringes with respect to the zero opd, namely the piston phase. By definition, the piston phase shift is the average differential phase across the apertures, as illustrated in Fig. 1 and described in Eq. (1), which can be rewritten as ${\mathit{\phi }}_{\mathrm{12}}^{\mathit{p}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}{\mathrm{\int }}_{\mathrm{\Sigma }}\mathrm{\left[}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathit{,t}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathit{,t}\mathrm{\left)}\mathrm{\right]}\mathrm{d\Sigma }\mathit{,}$(12)following the formalism of Buscher et al. (2008, see Eq. (6)), where dΣ is an elemental within the aperture Σ.

As first brought up by these authors, the practical estimate of this piston phase is performed by taking the argument of the complex phasor averaged over the pupils, that is $\begin{array}{}\mathrm{􏽦}\\ {\mathit{\phi }}_{\mathrm{12}}^{\mathit{p}}\end{array}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\arg \left({\mathrm{\int }}_{\mathrm{\Sigma }}{\mathrm{e}}^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathit{,t}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathit{,t}\mathrm{\left)}\mathrm{\right)}}\mathrm{d\Sigma }\right)$(13)in the multimode case, and $\begin{array}{}\mathrm{􏽦}\\ {\mathit{\phi }}_{\mathrm{12}}^{\mathit{p}}\end{array}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\arg \left({\mathrm{\int }}_{\mathrm{\Sigma }}{\mathrm{e}}^{\mathit{i}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{r}\mathit{,t}\mathrm{\right)}}\mathrm{d\Sigma }{\mathrm{\int }}_{\mathrm{\Sigma }}{\mathrm{e}}^{\mathrm{-}\mathit{i}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{\left(}{r}\mathit{,t}\mathrm{\right)}}\mathrm{d\Sigma }\right)$(14)in the singlemode one.

By comparing Eq. (12) to Eqs. (13, 14) we can see that the true piston phase and the estimated one are literally not the same, unless when there is no phase fluctuation (i.e. no turbulence or a perfect correction of the wavefronts) across the apertures.

When considering a small amount of fluctuations δφ₁₂(r,t) such as , a Taylor expansion of Eqs. (13, 14) can show that and will then deviate roughly as ^∫_Σ [ δφ₁₂(r,t) ] ³dΣ (Buscher et al. 2008). But if these fluctuations are high (typically above 1 radian), there can be strong differences between the piston phase and the argument of the complex phasor. More precisely, strong divergences are occurring when the complex phasor has a very small amplitude and eventually crosses the origin of the complex plane. Then rapid phase jumps of the argument of the phasor can be experienced, whereas these jumps are not seen in the piston phase. As a consequence the correction performed by the fringe tracker may be highly wrong and fringes potentially lost, especially if the science camera is working at a different wavelength than that of the fringe tracker (Buscher et al. 2008), as is the case e.g. for the FINITO instrument (fringe tracking in H-band and correction in K-band, Gai et al. 2004; Le Bouquin et al. 2008).

It is therefore important to know the probability of phase jumps to occur in order to ensure an observational/instrumental context in which these events are avoided as much as possible. So far, the conclusions about this point are not clear: if Tubbs (2005) implied that spatial filtering could be the source of these anomalies, Buscher et al. (2008) on the contrary argues that AO correction and most of all spatial filtering help to reduce these singularities. All the previous analysis were however based only on simulations. We propose here a theoretical analysis of this phenomenon. We emphasize that our analysis focuses on the rate of occurrence of phase jumps, as defined above. We do not study the consequences of these phase jumps in terms of an effective loss of fringe tracking. Such a causality will depend on the system (e.g. single wavelength vs. multiple spectral channels methods) used to practically estimate and correct the opd. Considering that a phase jump would systematically induce a failure in the fringe tracking system therefore corresponds to the worst case scenario.

That the coherent flux drops to zero because of turbulent phase fluctuation depends on whether the interferometric transfer function  |T₁₂|  for multimode systems or the interferometric coupling coefficient  |ρ₁₂|  in the singlemode case, drops to zero. We thus aim here to establish the probability density of these quantities, and study how likely it is for them to take very low values. Canales & Cagigal (1999) have studied the distribution of speckle statistics in presence of partial AO correction for mono-pupil telescopes. They have shown that the density probability dp(I) of the intensity I at the center of the image – that is by definition, the instantaneous Strehl ratio – follows a Rician statistics of the form $\mathit{dp}\mathrm{\left(}\mathit{I}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}{\mathit{\sigma }}^{\mathrm{2}}}\exp \left(\mathrm{-}\frac{\mathit{I}\mathrm{+}{\mathit{a}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}^{\mathrm{2}}}\right){\mathit{I}}_{\mathrm{0}}\left(\mathrm{-}\frac{\mathit{a}\sqrt{\mathit{I}}}{{\mathit{\sigma }}^{\mathrm{2}}}\right)\mathit{,}$(15)where a² and σ² depend on the first and second order moments of the real and imaginary part of the complex phasor describing the AO-corrected turbulent wavefront. Note that this analytical definition of the density probability implicitly assumes a time-constant r₀ to characterize the turbulence. This hypothesis is the limitation of our model because r₀ may actually vary on time scales of minutes and shorter, which leads to brief episodes of very small r₀ that would trip up AO systems and lead to a very low Strehl ratio for a brief period of time. However, modelling this effect requires more complex and heavy simulations of partially AO-corrected turbulence, which is beyond the scope of this paper.

We emphasized in Sect. 2 that  |T₁₂| ² and  |ρ₁₂| ² represent the instantaneous interferometric Strehl ratio for the multimode and singlemode cases, respectively. Therefore, by straightforward analogy with an interferometric pupil instead of a monolithic one, we can show that I =  |T₁₂| ² and I =  |ρ₁₂| ² are also following a Rician distribution. The parameters a² and σ² relative to both instantaneous interferometric Strehl ratio can be directly derived from the formalism of the previous section, and their expressions are given in Table B.1 of Appendix B.

Figure 6 shows the density probability of  |T₁₂| ² and  |ρ₁₂| ² for various levels of AO correction and turbulence strength.

High-order AO correction: first, we can see that as soon as moderate AO correction is applied – roughly a few tens of modes or a Strehl ratio  ≳ 0.3, the probability density displays a bell shape, which becomes narrower with a higher AO correction, and the probability that  |T₁₂|  or  |ρ₁₂|  goes to very low values is null (typically, for n_z = 15 and D/r0 = 8, the probability for the interferometric instantaneous Strehl ratio to be  < 0.01 is  < 10^-9, and  < 0.05 is  < 0.02%), emphasizing again the relevance of associating high-performance AO systems to fringe tracking devices, whether the wavefront is spatially filtered or not. We note that the intensity mean value (⟨ I ⟩  = 2σ² + a²) is systematically slightly higher in the singlemode case and the dispersion () slightly smaller. The difference is not critical however.

Low-order AO correction: for low-order AO such as tip-tilt correction only, the situation is fairly different. The shape of the distribution has changed, peaking at zero for strong turbulence with high values of D/r₀ ≳ 5. The probability to have a very low value of the intensity is thus significant, and events such as phase jumps are likely to happen. In other words, tip-tilt correction is an insufficient order correction to exclude these events.

Figure 7 displays the probability P(I) for the instantaneous interferometric Strehl ratio to reach values close to zero (i.e. P(I < ϵ) with ϵ very small) for different strengths of the atmospheric turbulence. We can see that for strong turbulence (D/r₀ ≳ 5), the probability P(I) < ϵ is always higher in the singlemode case. In these cases we expect phase jumps to occur more frequently with the singlemode fringe tracker, because the likelihood that two speckles are simultaneously entering the singlemode fibers of the two telescopes is weak. However, this does not mean that phase jumps cannot occur often with multimode fringe tracker too. As an example the probability to have instantaneous interferometric Strehl lower than 0.01 in the multimode case is  ~78% for D/r₀ = 8, whereas it is 100% in the singlemode. In other words, regardless of the multimode versus singlemode issue, one should never consider to perform fringe tracking with big telescope apertures associated with solely the tip-tilt correction.

For weak turbulence (roughly D/r₀ ≲ 5, Fig. 6, bottom middle and right), the distribution starts to look like a bell shape again, but this time with a slight probability to have a zero intensity. But as shown in Fig. 7 this probability is now higher in the multimode case, because only a few and big speckles are present in the images. Indeed, this case is the one treated by Buscher et al. (2008, see Fig. 4, with D/r₀ = 4 in his simulations, and we come to the same conclusion as he: in these conditions, spatial filtering lets the number of phase jump events decrease. We emphasize here though that this statement is true only for cases of moderate turbulence strength.

Application to the VLTI: We recall that at Paranal the average r₀ is  ~1 m in the K-band, which gives turbulence strengths of D/r₀ ~ 8 and D/r₀ ~ 1.8 for the UTs and ATs, respectively. The UTs come with high-order Adaptive Optic systems (MACAO, Arsenault et al. 2004), therefore one can equally choose multimode or singlemode fringe tracking schemes because (i) phase jumps are unlikely to occur in any case; and (ii) multimode and singlemode fringe trackers will provide the same robustness to eventual phase jumps. For ATs where tip-tilt correction is currently provided, the likelihood to endure phase jumps is also very low. But to maximize the stability of their fringe tracking system, it still seems appropriate to accompany the ATs with singlemode spatial filtering devices because the probability to undergo these phase jumps remains roughly 10 times lower (see Fig. 7, right). Alternatively, providing higher order AO correction to ATs will fix the phase jump issue.

4.3. Phase referencing – astrometry

The interest of phase referencing instruments such as PRIMA, which was recently installed on the VLTI (Delplancke et al. 2000), is (i) to provide sub-microarcsecond precision astrometry, allowing e.g. the detection of the presence of a faint companion (extrasolar planet) around the central star (Launhardt et al. 2008); and (ii) to drastically increase the limiting magnitude of the interferometer by locking the fringes of the (possibly faint) science object on a simultaneously observed bright off-axis reference star whose phase is used as reference (Sahlmann et al. 2008). This method requires that the nearby reference star is close enough – typically in the isoplanatic patch – in order to assume that both the wavefront of the source and the reference are identically perturbed by the atmospheric turbulence. Strictly speaking, this assumption is not true because the optical paths of the two stars are different, crossing different part of the atmosphere in the turbulent layers. As a result, the loss of correlation between the two wavefronts will lead to an atmospheric noise on the astrometric phase, which will lower the ultimate performance of this method.

We call respectively φ_s, and , the phase of the astrophysical target and the phase estimated from the off-axis reference source, located at an angular distance Δα of the science object, and where we assume for the sake of simplicity a single turbulent layer located at an height h from the ground. The astrometric phase is then simply defined by ${\mathrm{\Delta }}_{\mathit{\phi }}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{=}{\mathit{\phi }}_{\mathit{s}}\mathrm{-}{\mathit{\phi }}_{\mathit{r}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathit{,}$(16)and its associated error writes ${\mathit{\sigma }}_{{\mathrm{\Delta }}_{\mathit{\phi }}^{\mathrm{\Delta }\mathit{\alpha h}}}^{\mathrm{2}}\mathrm{=}\mathrm{2}{\mathit{\sigma }}_{\mathit{\phi }}^{\mathrm{2}}\mathrm{-}\mathrm{2}\mathrm{cov}\mathrm{\left(}{\mathit{\phi }}_{\mathit{s}}\mathit{,}{\mathit{\phi }}_{\mathit{r}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}\mathit{.}$(17)Equation (17) tells us that the noise of the astrometric phase increases as long as the correlation between the two turbulent wavefronts decreases, that is as long as the spatial distance Δαh between both turbulent phase screens is getting higher. The quantitative effect of the separation between the star and its reference on the interferometric narrow-angle astrometric error has already been studied by Shao & Colavita (1992), and we refer the readers to their paper for more details. In our analysis, we focus on the relative performance of astrometry between the multimode and singlemode scheme, studying how spatial filtering of the wavefronts will impact on the astrometric error, as developed in Appendix C.

For sake of simplicity we have assumed in the following one single turbulent layer located at h = 10 km as it is the strongest layer at the Cerro Paranal site (Masciadri et al. 1999). There is also a strong layer very close to the ground at around 20 m (Martin et al. 2000) but that only marginally contributes to the phase decorrelation, because the linear distance at stake is 500 times smaller than the one at the high altitude layer. Figure 8 (left) shows the evolution of the astrometric error as a function of the angular distance between the source and the reference for both the singlemode and multimode scheme.

Large field of view: when the reference source is farther away than about  ≃1′′ from the astrophysical object of interest, we can see that spatial filtering allows us to increase the precision of the astrometric phase. The improvement is better when the AO correction is low, reaching a factor of  ~4 without AO correction, as emphasized by Fig. 8 (right). This argues in favor of using spatial-filtering elements in the design of astrometric instruments with a large field of view (FOV), as was chosen for PRIMA (FOV ≃ 30′′). Looking at more details, one can notice that the phase tends asymptotically toward a plateau, from an angular separation of  ~10′′ that roughly corresponds to the isoplanatic angle of Paranal in the K-band⁵. Indeed, from this angle wavefronts can be considered as uncorrelated and the astrometric error converges to , which does not depend on the separation Δα any longer. As a consequence in this regime (i.e. Δα ≳ 10′′) the astrometric phase error ratio can be approximated by the speckle phase error ratio, which we already discussed in Sect. 3.3.

Narrow field of view: on the contrary, when wavefronts are still strongly correlated (Δα ≲ 1′′), the performance of singlemode astrometry is slightly poorer (by a factor  ~1 to  ~1.5) than that of multimode astrometry, the error of the latter decreasing faster as the separation between the astrophysical source and the reference shrinks to zero. By smoothing the turbulent wavefronts across the apertures, spatial filtering is indeed lowering the effect of the strong correlation between the wavefronts. This situation concerns the cases for which the reference is very close to the star, like GRAVITY, the second generation of astrometric instruments of the VLTI (Gillessen et al. 2006, FOV ≃ 2′′).

5. Summary and conclusions

We provided a theoretical formalism that allows us to derive the error of the interferometric phase both for singlemode and multimode interferometry. From these derivations, we demonstrated that:

• Contrarily to a widespread idea, losing flux by injecting the lightinto singlemode spatial filters is not a performance killer forestimating the phase. Indeed singlemode interferometryprovides a better performance than that of multimodeinterferometry, unless the interferometer is working in thedetector noise regime (faint sources). Then multimodeinterferometry is slightly better, providing a phase error smallerby a factorρ₀ ≃ 0.8, which is the maximum fraction of flux that can be injected in singlemode devices.

• In cases of bright source observations, spatial filtering is shown to be very efficient, especially when the AO correction of the turbulent wavefronts is poor or absent. In these situations, the precision of the singlemode interferometric phase is better than that of the multimode one by a factor of 2 and more when the Strehl is below 10%.

• Singlemode interferometry also proves to be more robust to the turbulence of both locking and coherently integrating the fringes and providing a better astrometric precision when using phase referencing techniques, except for narrow fields of view (FOV ≲ 1′′).

In conclusion, from a theoretical point of view and contrarily to a widespread opinion, singlemode fringe tracking should be seriously considered as an advantageous technical solution. Furthermore, the astronomers should realize that the many gains of singlemode interferometry (flexibility of the solutions, robustness to the alignment, less optical elements,...) may significantly compensate a modest and limited loss of performance in the faint sources case compared to multimode interferometry solutions. This is all the more true because the detectors should soon evolve to reach photon counting capability.

---

Acknowledgments

The authors are grateful to the anonymous referee, whose careful and thorough review of the text and theoretical formalism helped them improve the papers clarity and quality considerably.

References

Appendix A: Computation of the interferometric phase error

A.1. General formalism

We here recall Goodman’s approach (Goodman 1985), which is based on a continuous model of detection process, where the recorded signal (i.e. the interferogram) can be represented as $\mathit{d}\mathrm{\left(}\mathit{x,y}\mathrm{\right)}\mathrm{=}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{K}}\mathit{\delta }\mathrm{(}\mathit{x}\mathrm{-}{\mathit{x}}_{\mathit{k}}\mathit{,y}\mathrm{-}{\mathit{y}}_{\mathit{k}}\mathrm{)}\mathit{,}$(A.1)and its Fourier transform as $\mathit{D}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{=}\mathit{D}\mathrm{\left(}{\mathit{f}}_{\mathit{x}}\mathit{,}{\mathit{f}}_{\mathit{y}}\mathrm{\right)}\mathrm{=}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{K}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}\mathit{i\pi }\mathrm{(}{\mathit{f}}_{\mathit{x}}{\mathit{x}}_{\mathit{k}}\mathrm{+}{\mathit{f}}_{\mathit{y}}{\mathit{y}}_{\mathit{k}}\mathrm{)}}\mathit{,}$(A.2)where the position (x_n) and the number of photoevents (per time unit) (K) are statistical processes with probability laws depending on the intensity distribution I(x). To take also into account the detector noise, one needs to add Gaussian additive noise (ϵ) to the previous equation (Tatulli et al. 2004; Tatulli & Chelli 2005): $\mathit{Q}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{=}\mathit{D}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}\mathit{ϵ}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{=}\sum _{\mathit{k}\mathrm{=}\mathrm{1}}^{\mathit{K}}\exp {\mathit{o}}^{\mathrm{-}\mathrm{2}\mathit{i\pi }\mathrm{(}{\mathit{f}}_{\mathit{x}}{\mathit{x}}_{\mathit{k}}\mathrm{+}{\mathit{f}}_{\mathit{y}}{\mathit{y}}_{\mathit{k}}\mathrm{)}}\mathrm{+}\mathit{ϵ}\mathrm{\left(}{f}\mathrm{\right)}\mathit{.}$(A.3)Note that this formalism is also valid for a temporal coding of the interferogram, which is replacing the spatial position (x_k,y_k) by the (1-D) temporal sampling t, and similarly the spatial frequency f by the temporal one ν.

As Q represents the spectrum of the interferogram, the phase of the spectrum is merely the argument of this estimator: $\mathit{\phi }\mathrm{=}\mathrm{atan}\left(\mathrm{Im}\mathrm{\left(}\mathit{Q}\mathrm{\right)}\mathit{,}\mathrm{Re}\mathrm{\left(}\mathit{Q}\mathrm{\right)}\right)\mathit{,}$(A.4)In this framework, Chelli (1989) has shown that in first approximation (i.e. small noise error), the variance of the phase can be expressed as ${\mathit{\sigma }}_{\mathit{\phi }}^{\mathrm{2}}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathit{E}\mathrm{\left(}\mathrm{\right|}\mathit{Q}{\mathrm{|}}^{\mathrm{2}}\mathrm{)}\mathrm{-}\mathrm{Re}\mathrm{[}\mathit{E}\mathrm{\left(}{\mathit{Q}}^{\mathrm{2}}\mathrm{\right)}\mathrm{]}}{\mathrm{\left[}\mathit{E}\mathrm{\right(}\mathit{Q}\mathrm{)}{\mathrm{]}}^{\mathrm{2}}}\mathit{,}$(A.5)then using Goodman’s formalism described above, one can show that (Chelli 1989; Tatulli et al. 2004) $\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathrm{\right|}\mathit{Q}{\mathrm{|}}^{\mathrm{2}}\mathrm{)}& \mathrm{=}& \mathrm{⟨}{\overline{\mathit{K}}}^{\mathrm{2}}\mathrm{\left|}\mathit{i}\mathrm{\right(}{f}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}\mathrm{+}\overline{\mathit{K}}\mathrm{+}{\mathit{N}}_{\mathrm{pix}}{\mathit{\sigma }}_{\det }^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}\\ \mathit{E}\mathrm{\left(}{\mathit{Q}}^{\mathrm{2}}\mathrm{\right)}& \mathrm{=}& \mathrm{⟨}{\overline{\mathit{K}}}^{\mathrm{2}}{\mathit{i}}^{\mathrm{2}}\mathrm{\left(}{f}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\\ \mathit{E}\mathrm{\left(}\mathit{Q}\mathrm{\right)}& \mathrm{=}& \mathrm{⟨}\overline{\mathit{K}}\mathit{i}\mathrm{\left(}{f}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathit{,}\end{array}$where is the average number of photoevents per time unit, σ_det is the detector noise and N_pix is the number of pixel to sample the signal. i(f) is the normalized spectrum of the interferogram (i.e. i(0) = 1).

 ⟨  ⟩ _Φ denotes the expected value with respect to the atmosphere. How the previous expressions are unfolding depends on whether we deal with multimode or singlemode interferometry.

A.2. Useful definitions

In this section we introduce the concepts and notations that we will use in our formalism to derive the expression of the noise of the interferometric phase. Let us consider (see also Fig. 1 in the paper)

• an interferometer made of two telescopes described by their pupil functions P₁(r), P₂(r) and their associated transmission , . Note that with this a definition P₁ and P₂ are centered respectively at the position r₁ = λf₁ and r₂ = λf₂, and we introduce the quantity f₁₂ = f₂ − f₁ as the baseline frequency of the interferometer.

• and the partially AO-corrected atmospheric turbulent phase screens. We also introduce the structure function of the residual phases (assuming the same level of AO correction for each telescope) together with , its associated residual phase variance (see also Appendix D).

• the incoming wavefront of the source Ψ(r). By definition we have  |Ψ(0)| ² = N, where N is the number of photons per surface unit and per time unit emitted by the source, and  [Ψ(r)Ψ ^∗ (r + λf)] /|Ψ(0)| ² = V(f) the visibility of the source at the spatial frequency f.

We emphasize that the following derivations will be done assuming monochromatic interferograms at the wavelength λ. Indeed, taking into account a non-null spectral bandwidth δλ is equivalent to considering a monochromatic interferogram at the effective wavelength λ₀ of the filter, modulated in amplitude by an envelope, whose width is fixed by the spectral coherence length . Moreover, this effect is independent of the multimode or singlemode nature of the interferogram.

A.3. Phase noise in multimode interferometry

Below we derive the estimator of the coherent flux of the interferogram, and subsequently the error of the interferometric phase in the multimode case. We provide two formalisms that respectively describe both beam recombination schemes: in the image plane and in the pupil plane.

A.3.1. Image plane recombination

Image plane recombination is well suited to describe multiaxial interferometric schemes where the fringe pattern is spatially coded on the detector, such as for the GI2T and VEGA/CHARA instruments. Note that co-axial (i.e. temporal) coding can also be performed in the image plane, but in practice – to our knowledge – no interferometric instruments are using this technique.

In this case, the spatial distribution of the complex amplitude of the radiation field E(r) can be written as $\mathit{E}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{=}\sqrt{{\mathit{t}}_{\mathrm{1}}}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\exp }^{\mathit{i}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}}\mathrm{\Psi }\mathrm{\left(}{r}\mathrm{\right)}\mathrm{+}\sqrt{{\mathit{t}}_{\mathrm{2}}}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\exp }^{\mathit{i}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}}\mathrm{\Psi }\mathrm{\left(}{r}\mathrm{\right)}\mathit{.}$(A.9)By rules of diffraction theory, the Fourier Transform of the interference pattern formed in the image plane is the autocorrelation of the complex amplitude E(r), namely $\begin{array}{ccc}\mathit{I}\mathrm{\left(}{f}\mathrm{\right)}& \mathrm{=}& \int \mathit{E}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{E}}^{\mathrm{\ast }}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{d}{r}\\ & \mathrm{=}& {\mathit{t}}_{\mathrm{1}}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{\Psi }\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{\Psi }}^{\mathrm{\ast }}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{d}{r}\\ & \mathrm{+}& {\mathit{t}}_{\mathrm{2}}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{\Psi }\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{\Psi }}^{\mathrm{\ast }}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{d}{r}\\ & \mathrm{+}& \sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{\Psi }\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{\Psi }}^{\mathrm{\ast }}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{d}{r}\\ & \mathrm{+}& \sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}{\mathrm{\Psi }}^{\mathrm{\ast }}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{\Psi }\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathit{.}\end{array}$(A.10)Introducing  | Ψ(0) | ² = N in the above equation enables us to make the complex visibility function appear: $\mathit{I}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{=}\mathit{N}{\mathit{t}}_{\mathrm{1}}\mathit{V}\mathrm{\left(}{f}\mathrm{\right)}{\mathit{S}}_{\mathrm{1}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}\mathit{N}{\mathit{t}}_{\mathrm{2}}\mathit{V}\mathrm{\left(}{f}\mathrm{\right)}{\mathit{S}}_{\mathrm{2}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}\mathit{V}\mathrm{\left(}{f}\mathrm{\right)}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}\mathit{V}\mathrm{\left(}{f}\mathrm{\right)}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\ast }}\mathrm{(}\mathrm{-}{f}\mathrm{)}\mathit{,}$(A.11)with S₁(f) and S₂(f) the so-called photometric peaks resulting from the autocorrelation of each corrugated pupil, and S₁₂(f) the interferometric peak arising from the cross-correlation between both pupils: $\begin{array}{ccc}{\mathit{S}}_{\mathit{i}}\mathrm{\left(}\mathit{f}\mathrm{\right)}\mathrm{=}\mathrm{\int }{\mathit{P}}_{\mathit{i}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathit{i}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathit{i}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathit{i}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}\mathit{,} \mathrm{(}\mathit{i}\mathrm{=}\mathrm{[}\mathrm{1}\mathit{,}\mathrm{2}\mathrm{\left]}\mathrm{\right)}& & \end{array}$(A.12)$\begin{array}{ccc}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}\mathit{f}\mathrm{\right)}\mathrm{=}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}\mathit{.}& & \end{array}$(A.13)From Eq. (A.11), we can straightforwardly derive the number of photoevents: $\overline{\mathit{K}}\mathrm{=}\mathit{I}\mathrm{\left(}\mathbf{0}\mathrm{\right)}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}\mathit{,}$(A.14)which is turbulence-independent (neglecting scintillation). Σ_P is the collecting area of a single telescope: ${\mathrm{\Sigma }}_{\mathit{P}}\mathrm{=}\mathrm{\int }\mathrm{\left[}{\mathit{P}}_{\mathrm{1}}\mathrm{\right(}{r}\mathrm{)}{\mathrm{]}}^{\mathrm{2}}\mathrm{d}{r}\mathrm{=}\mathrm{\int }\mathrm{[}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{]}}^{\mathrm{2}}\mathrm{d}{r}\mathit{,}$(A.15)assuming for the sake of simplicity that both pupils are identical.

Equation (A.11) also shows that multimode interferometry with image plane recombination continuously transmits the whole spatial frequencies f. The complex visibility V₁₂ = V(f₁₂) is then derived from the estimate of the coherent flux at the baseline frequency of the interferometer , that is ${\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\mathrm{=}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}\mathit{,}$(A.16)and the normalized spectrum of the interferogram takes the form $\mathit{i}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}\mathrm{=}\frac{\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}}{\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}{\mathrm{\Sigma }}_{\mathit{P}}}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}\mathrm{=}\frac{\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}}{\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}}{\mathit{T}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}\mathit{,}$(A.17)where T₁₂(f) = S₁₂(f)/Σ_P is the normalized interferometric transfer function, such as  |T₁₂(f₁₂) |  = 1. As a consequence, by analogy with the definition of the Strehl ratio for a single telescope,  |T₁₂(f₁₂) | ² can be seen as the instantaneous multimode interferometric Strehl ratio.

A.3.2. Pupil plane recombination

In this scheme, fringes are formed in the pupil plane by means of geometrical operations of the entrance pupil (Chelli & Mariotti 1986). This formalism is usually well suited to describe co-axial recombination with temporal coding where the pupils are superimposed at the beam splitter, hence the fringes formed in the pupil plane, as in the IOTA instrument. In practice, the fringes are form by translating one pupil over the other (translation of vector ), and by introducing an optical path delay (θ₁(t) and θ₂(t)) on each beam (see e.g. Buscher 1988, Appendix A2) to modulate the fringe pattern. In this case, the complex amplitude of the superimposed electric fields writes $\mathit{E}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{=}\sqrt{{\mathit{t}}_{\mathrm{1}}}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\exp }^{\mathit{i}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}}\mathrm{\Psi }\mathrm{\left(}{r}\mathrm{\right)}{\exp }^{\mathit{i}{\mathit{\theta }}_{\mathrm{1}}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathrm{+}\sqrt{{\mathit{t}}_{\mathrm{2}}}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathit{i}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}}\mathrm{\Psi }\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathit{i}{\mathit{\theta }}_{\mathrm{2}}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathit{.}$(A.18)The fringe pattern I(t) is then formed by focusing the light on one single pixel of the detector. Again, by rules of image formation, it writes as $\mathit{I}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{\int }\mathrm{\left|}\mathit{E}\mathrm{\right(}{r}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}\mathrm{d}{r}\mathit{,}$(A.19)which, by making use of the expression of E(r) rewrites as $\begin{array}{ccc}\mathit{I}\mathrm{\left(}\mathit{t}\mathrm{\right)}& \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}\mathrm{+}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\exp }^{\mathit{i}{\mathrm{\Delta }}^{\mathrm{12}}\mathit{\theta }\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}}\mathrm{\Psi }\mathrm{\left(}{r}\mathrm{\right)}{\mathrm{\Psi }}^{\mathrm{\ast }}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{d}{r}\mathrm{+}\mathit{CC}\\ & \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}\mathrm{+}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}\mathit{V}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\exp }^{\mathit{i}{\mathrm{\Delta }}^{\mathrm{12}}\mathit{\theta }\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathrm{+}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}\mathit{V}\mathrm{(}\mathrm{-}{f}\mathrm{12}\mathrm{)}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\ast }}\mathrm{(}\mathrm{-}{f}\mathrm{12}\mathrm{)}{\exp }^{\mathrm{-}\mathit{i}{\mathrm{\Delta }}^{\mathrm{12}}\mathit{\theta }\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathit{,}\end{array}$(A.20)where Δ¹²θ(t) = θ₁(t) − θ₂(t) is the phase delay between the two beams that sample the fringe pattern. In the Fourier plane, the interferometric equation takes the form $\mathit{I}\mathrm{\left(}\mathit{\nu }\mathrm{\right)}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}\mathrm{+}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}\mathit{V}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathit{\delta }}_{{\mathit{\nu }}_{\mathrm{12}}}\mathrm{+}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}^{\mathrm{\ast }}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\ast }}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathit{\delta }}_{\mathrm{-}{\mathit{\nu }}_{\mathrm{12}}}\mathit{,}$(A.21)where δ_ν is the Dirac function, and ν₁₂ is the temporal frequency at which the interferogram is modulated, that is the frequency of the moving piezoelectric mirror.

Note that at the difference of the image plane recombination, pupil plane interferometry transmits only the frequency baseline . Nonetheless, the coherent flux, estimated in the Fourier plane at the frequency mirror (ν = ν₁₂) takes the same form as in the image plane case, namely: ${\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\mathrm{=}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}\mathit{,}$(A.22)and similarly the total number of detected photoevents is $\overline{\mathit{K}}\mathrm{=}\mathit{I}\mathrm{\left(}\mathbf{0}\mathrm{\right)}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}\mathit{,}$(A.23)and the normalized spectrum of the interferogram at the frequency ν₁₂ is $\mathit{i}\mathrm{\left(}{\mathit{\nu }}_{\mathrm{12}}\mathrm{\right)}\mathrm{=}\frac{\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}}{\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}{\mathrm{\Sigma }}_{\mathit{P}}}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}\mathit{.}$(A.24)

A.3.3. The interferometric phase error

In the above sections, we showed that independently of the chosen recombination scheme, the total flux, the coherent flux, and consequently the normalized spectrum of the interferogram can take the form of Eqs. (A.14–A.17), respectively. Using these expressions, Eqs. (A.6–A.8) can be rewritten as $\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathrm{\right|}\mathit{Q}{\mathrm{|}}^{\mathrm{2}}\mathrm{)}& \mathrm{=}& {\mathit{N}}^{\mathrm{2}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}\mathrm{|}{\mathit{V}}_{\mathrm{12}}{\mathrm{|}}^{\mathrm{2}}\mathrm{⟨}\mathrm{|}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}\mathrm{+}{\mathit{N}}_{\mathrm{pix}}{\mathit{\sigma }}_{\det }^{\mathrm{2}}\end{array}$(A.25)$\begin{array}{ccc}\mathit{E}\mathrm{\left(}{\mathit{Q}}^{\mathrm{2}}\mathrm{\right)}& \mathrm{=}& {\mathit{N}}^{\mathrm{2}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}{\mathit{V}}_{\mathrm{12}}^{\mathrm{2}}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}^{\mathrm{2}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\end{array}$(A.26)$\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathit{Q}\mathrm{\right)}& \mathrm{=}& \mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathit{\phi }}\mathit{.}\end{array}$(A.27)NB: Chelli (1989) has shown that the error of the phase does not depend on the object phase, hence one can consider in the following that the object is centro-symmetric, that is V₁₂ =  |V₁₂|  = Re [ V₁₂ ] . What remains now is to derive from the above equations the moments of the interferometric transfer function S₁₂(f₁₂): $\begin{array}{ccc}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\\ & \mathrm{=}& \int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}\mathrm{\left|}\mathrm{\right(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}}\mathrm{d}{r}\\ & \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}{\exp }^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\end{array}$(A.28)$\begin{array}{ccc}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}^{\mathrm{2}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & \mathrm{=}& {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}\mathrm{\left|}\mathrm{\right(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{knowingthat} \mathrm{⟨}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{=}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}\mathit{,} \mathrm{itcomes}\mathit{,}\\ & \mathrm{=}& {\exp }^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{changingaxesreference:} {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathit{,} {\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)} \mathrm{\to } \mathit{P}\mathrm{\left(}{r}\mathrm{\right)} \mathrm{centeredon}\mathrm{0}\mathit{,}\\ & \mathrm{=}& {\exp }^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{\int }\left[\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\right]\left[\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\right]\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\end{array}$(A.29)$\begin{array}{ccc}\mathrm{⟨}\mathrm{\left|}{\mathit{S}}_{\mathrm{12}}\mathrm{\right(}{f}\mathrm{12}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & \mathrm{=}& \int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}\mathrm{\left|}\mathrm{\right(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & \mathrm{=}& \int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{changingaxesreference:}\\ & \mathrm{=}& \mathrm{\int }\left[\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\right]\left[\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\right]\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathit{.}\end{array}$(A.30)Putting Eqs. (A.28, A.29, A.30) into Eqs. (A.25, A.26, A.27), we finally find that the variance of the phase can be written as the quadratic sum of three terms, corresponding to the detector, photon, and atmospheric regimes respectively, with

$\begin{array}{ccc}{\mathit{\sigma }}_{\mathrm{de}{\mathrm{t}}_{\mathit{\phi }}}^{\mathrm{2}}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{{\mathit{N}}_{\mathrm{pix}}{\mathit{\sigma }}_{\det }^{\mathrm{2}}}{{\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}{\mathit{N}}^{\mathrm{2}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}\mathrm{|}{\mathit{V}}_{\mathrm{12}}{\mathrm{|}}^{\mathrm{2}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}}\end{array}$(A.31)$\begin{array}{ccc}{\mathit{\sigma }}_{{\mathrm{phot}}_{\mathit{\phi }}}^{\mathrm{2}}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}}{{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}\mathrm{|}{\mathit{V}}_{\mathrm{12}}{\mathrm{|}}^{\mathrm{2}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}}\end{array}$(A.32)$\begin{array}{ccc}{\mathit{\sigma }}_{{\mathrm{atm}}_{\mathit{\phi }}}^{\mathrm{2}}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }{\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathrm{e}}^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}}^{\mathrm{2}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{-}{\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{\int }{\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathrm{e}}^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}}^{\mathrm{2}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}^{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}}\mathit{.}\end{array}$(A.33)We note that the expression of the variance for the photon noise () is the generalization of the formula found by Vannier et al. (2006) (Eq. (14)), when the visibility is corrupted by the partially AO-corrected turbulent phase.

A.4. Phase noise in singlemode interferometry

Because of the spatial filtering specific properties of singlemode devices, the singlemode interferometric equation changes slightly with respect to that of the multimode, because we have to take into account the coupling efficiency that extenuates the number of coherent and incoherent photoevents. The coupling coefficients, for the photometric (ρ₁(V), ρ₂(V)), and interferometric (ρ¹²(V)) channels respectively write (Mège et al. 2003; Tatulli et al. 2004) $\begin{array}{ccc}{\mathit{\rho }}_{\mathit{i}}\mathrm{\left(}\mathit{V}\mathrm{\right)}& \mathrm{=}& {\mathit{\rho }}_{\mathrm{0}}\frac{\mathrm{(}\mathit{V}\mathrm{\ast }{\mathit{S}}_{\mathit{i}}\mathrm{)}\mathrm{\left(}\mathbf{0}\mathrm{\right)}}{\mathrm{\int }{\mathit{S}}_{\mathit{i}}^{\mathrm{0}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{d}{f}}\mathit{,} \mathrm{(}\mathit{i}\mathrm{=}\mathrm{[}\mathrm{1}\mathit{,}\mathrm{2}\mathrm{\left]}\mathrm{\right)}\\ & \mathrm{=}& {\mathit{\rho }}_{\mathrm{0}}\frac{\mathrm{\int }\mathit{V}\mathrm{\left(}\mathit{f}\mathrm{\right)}{\mathit{P}}_{\mathit{i}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathit{i}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathit{i}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathit{i}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{f}}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathit{i}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\end{array}$(A.34)$\begin{array}{ccc}{\mathit{\rho }}^{\mathrm{12}}\mathrm{\left(}\mathit{V}\mathrm{\right)}& \mathrm{=}& {\mathit{\rho }}_{\mathrm{0}}\frac{\mathrm{(}\mathit{V}\mathrm{\ast }{\mathit{S}}_{\mathrm{12}}\mathrm{)}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}}{\sqrt{\mathrm{\int }{\mathit{S}}_{\mathrm{1}}^{\mathrm{0}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{d}{f}\mathrm{\int }{\mathit{S}}_{\mathrm{2}}^{\mathrm{0}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{d}{f}}}\\ & \mathrm{=}& {\mathit{\rho }}_{\mathrm{0}}\frac{\mathrm{\int }\mathit{V}\mathrm{(}\mathit{f}\mathrm{-}{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathit{,} \\ & & \end{array}$where , are the photometric transfer functions of a perfect (atmosphere-free) interferometer, and ρ₀ the maximum coupling efficiency, shown to be  ~80% (Shaklan & Roddier 1988). And the singlemode interferometric equation writes: $\mathit{I}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{1}}{\mathit{\rho }}_{\mathrm{1}}\mathrm{\left(}\mathit{V}\mathrm{\right)}{\mathit{H}}_{\mathrm{1}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{2}}\mathrm{\left(}\mathit{V}\mathrm{\right)}{\mathit{H}}_{\mathrm{2}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}^{\mathrm{12}}\mathrm{\left(}\mathit{V}\mathrm{\right)}{\mathit{H}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}\mathrm{\left(}{\mathit{\rho }}^{\mathrm{12}}\mathrm{\right(}\mathit{V}\mathrm{\left)}{\mathrm{)}}^{\mathrm{\ast }}{\mathit{H}}_{\mathrm{12}}^{\mathrm{\ast }}\mathrm{\right(}\mathrm{-}{f}\mathrm{)}\mathit{,}$(A.37)where H₁(f), H₂(f) and H₁₂(f) are the normalized (i.e. H₁(0) = H₂(0) = 1, and H₁₂(f₁₂) = 1) singlemode photometric and interferometric transfer functions, the so-called carrying wave (Mège 2002), which are fixed by the geometry of the singlemode device and therefore independent of the atmosphere. As a result, the singlemode coherent flux is defined by ${\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}^{\mathrm{12}}\mathrm{\left(}\mathit{V}\mathrm{\right)}\mathit{.}$(A.38)

A.4.1. Compact sources case

For compact sources – which are those unresolved by the telescope, which is the most common case in interferometry – Eqs. (A.34) and (A.35) are taking the simplified following form: $\begin{array}{ccc}{\mathit{\rho }}_{\mathit{i}}\mathrm{\left(}\mathit{V}\mathrm{\right)}& \mathrm{=}& {\mathit{\rho }}_{\mathrm{0}}\frac{\mathrm{\int }{\mathit{P}}_{\mathit{i}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathit{i}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathit{i}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathit{i}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{f}}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathit{i}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\mathrm{=}{\mathit{\rho }}_{\mathrm{0}}{\mathrm{𝒮}}_{\mathit{i}}\end{array}$(A.39)$\begin{array}{ccc}{\mathit{\rho }}^{\mathrm{12}}\mathrm{\left(}\mathit{V}\mathrm{\right)}& \mathrm{=}& {\mathit{\rho }}_{\mathrm{0}}{\mathit{V}}_{\mathrm{12}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathrm{=}{\mathit{\rho }}_{\mathrm{0}}{\mathit{V}}_{\mathrm{12}}{\mathit{\rho }}_{\mathrm{12}}\mathit{,} \\ & & \end{array}$where is by definition the instantaneous Strehl ratio relative to the ith telescope, and with $\begin{array}{ccc}{\mathit{\rho }}_{\mathrm{12}}& \mathrm{=}& \frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathit{,}\end{array}$(A.42) |ρ₁₂| ² being by analogy the instantaneous singlemode interferometric Strehl ratio. The interferometric equation finally rewrites $\mathit{I}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{1}}{\mathit{\rho }}_{\mathrm{0}}{\mathrm{𝒮}}_{\mathrm{1}}{\mathit{H}}_{\mathrm{1}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{0}}{\mathrm{𝒮}}_{\mathrm{2}}{\mathit{H}}_{\mathrm{2}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}{\mathit{\rho }}_{\mathrm{12}}{\mathit{H}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}_{\mathrm{0}}{\mathit{V}}_{\mathrm{12}}^{\mathrm{\ast }}{\mathit{\rho }}_{\mathrm{12}}^{\mathrm{\ast }}{\mathit{H}}_{\mathrm{12}}^{\mathrm{\ast }}\mathrm{(}\mathrm{-}{f}\mathrm{)}\mathit{.}$(A.43)And the singlemode coherent flux at the spatial frequency takes the form: ${\mathit{F}}_{\mathrm{12}}^{\mathit{c}}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}{\mathit{\rho }}_{\mathrm{12}}\mathit{,}$(A.44)whereas the number of photoevents writes $\overline{\mathit{K}}\mathrm{=}\mathit{I}\mathrm{\left(}\mathbf{0}\mathrm{\right)}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{\rho }}_{\mathrm{0}}{\mathrm{𝒮}}_{\mathrm{1}}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{0}}{\mathrm{𝒮}}_{\mathrm{2}}\mathit{,}$(A.45)which depends on the turbulent atmosphere through the fluctuations of the instantaneous Strehl ratio and . And the normalized spectrum of the interferogram is $\mathit{i}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}\mathrm{=}\frac{\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}_{\mathrm{12}}{\mathit{V}}_{\mathrm{12}}}{{\mathit{t}}_{\mathrm{1}}{\mathrm{𝒮}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}{\mathrm{𝒮}}_{\mathrm{2}}}\mathrm{·}$(A.46)The generic equations of the phase noise thus rewrites $\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathrm{\right|}\mathit{Q}{\mathrm{|}}^{\mathrm{2}}\mathrm{)}& \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}{\mathit{N}}^{\mathrm{2}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{0}}^{\mathrm{2}}\mathrm{|}{\mathit{V}}_{\mathrm{12}}{\mathrm{|}}^{\mathrm{2}}\mathrm{⟨}\mathrm{|}{\mathit{\rho }}_{\mathrm{12}}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{1}}{\mathit{\rho }}_{\mathrm{0}}\mathrm{⟨}{\mathrm{𝒮}}_{\mathrm{1}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{+}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{0}}\mathrm{⟨}{\mathrm{𝒮}}_{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{+}{\mathit{N}}_{\mathrm{pix}}{\mathit{\sigma }}_{\det }^{\mathrm{2}}\end{array}$(A.47)$\begin{array}{ccc}\mathit{E}\mathrm{\left(}{\mathit{Q}}^{\mathrm{2}}\mathrm{\right)}& \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}{\mathit{N}}^{\mathrm{2}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{0}}^{\mathrm{2}}{\mathit{V}}_{\mathrm{12}}^{\mathrm{2}}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}\end{array}$(A.48)$\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathit{Q}\mathrm{\right)}& \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}_{\mathrm{0}}{\mathit{V}}_{\mathrm{12}}\mathit{.}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}{\mathrm{⟩}}_{\mathrm{\Phi }}\end{array}$(A.49)Now we need to compute the moments of the coupling coefficients: $\begin{array}{ccc}\mathrm{⟨}{\mathrm{𝒮}}_{\mathit{i}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \overline{{\mathrm{𝒮}}_{\mathit{i}}}\mathit{,} \mathrm{(}\mathrm{i}\mathrm{=}\mathrm{[}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{\left]}\mathrm{\right)}\mathit{,}\end{array}$(A.50)where is the long exposure Strehl ratio of the ith telescope. Assuming equivalent AO correction for both telescopes, we have $\begin{array}{ccc}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathrm{=}{\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathrm{=}{\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathit{,}\end{array}$(A.51)which shows that the long exposure interferometric Strehl ratio is equal to the coherent energy . $\begin{array}{ccc}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & \mathrm{=}& {\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{changeofvariables} {r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\to }{u}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\to }{u}\mathrm{\prime }\mathrm{:}\\ & \mathrm{=}& {\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\end{array}$(A.52)$\begin{array}{ccc}\mathrm{⟨}\mathrm{|}{\mathit{\rho }}_{\mathrm{12}}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & \mathrm{=}& \frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{changeofvariables} {r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\to }{u}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\to }{u}\mathrm{\prime }\mathrm{:}\\ & \mathrm{=}& \frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\mathrm{·}\end{array}$(A.53)Putting Eqs. (A.50, A.51, A.52, A.53) into Eqs. (A.47, A.48, A.49), we find again that the variance of the phase can be written as the quadratic sum of three terms, corresponding to the detector, photon, and atmospheric regimes respectively, with $\begin{array}{ccc}{\mathit{\sigma }}_{\mathrm{de}{\mathrm{t}}_{\mathit{\phi }}}^{\mathrm{2}}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{{\mathit{N}}_{\mathrm{pix}}{\mathit{\sigma }}_{\det }^{\mathrm{2}}}{{\mathit{\rho }}_{\mathrm{0}}^{\mathrm{2}}{\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}{\mathit{N}}^{\mathrm{2}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}\mathrm{|}{\mathit{V}}_{\mathrm{12}}{\mathrm{|}}^{\mathrm{2}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}}\end{array}$(A.54)$\begin{array}{ccc}{\mathit{\sigma }}_{\mathrm{pho}{\mathrm{t}}_{\mathit{\phi }}}^{\mathrm{2}}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{(}{\mathit{t}}_{\mathrm{1}}\mathrm{+}{\mathit{t}}_{\mathrm{2}}\mathrm{)}\overline{\mathrm{𝒮}}}{{\mathit{\rho }}_{\mathrm{0}}{\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}\mathrm{|}{\mathit{V}}_{\mathrm{12}}{\mathrm{|}}^{\mathrm{2}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}}\end{array}$(A.55)$\begin{array}{ccc}{\mathit{\sigma }}_{{\mathrm{atm}}_{\mathit{\phi }}}^{\mathrm{2}}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{{{\mathrm{[}}^{\mathrm{\int }}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathrm{e}}^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{{r}\mathrm{\prime }}^{\mathrm{]}}}^{\mathrm{2}}\mathrm{-}{\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}{{\mathrm{[}}^{\mathrm{\int }}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathrm{e}}^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{{r}\mathrm{\prime }}^{\mathrm{]}}}^{\mathrm{2}}}{{{\mathrm{[}}^{\mathrm{\int }}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{4}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}}\mathrm{·}\end{array}$(A.56)

Appendix B: Coherent flux dropouts - Rician density probability

By analogy with the study of Canales & Cagigal (1999) on the statistics of the instantaneous Strehl ratio in partial AO correction, the instantaneous interferometric Strehl ratio, namely I =  |T₁₂| ² and I =  |ρ₁₂| ² for multimode and singlemode interferometers respectively are shown to follow Rician density probability statistics of the form: $\mathit{dp}\mathrm{\left(}\mathit{I}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}{\mathit{\sigma }}^{\mathrm{2}}}\exp \left(\mathrm{-}\frac{\mathit{I}\mathrm{+}{\mathit{a}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}^{\mathrm{2}}}\right){\mathit{I}}_{\mathrm{0}}\left(\mathrm{-}\frac{\mathit{a}\sqrt{\mathit{I}}}{{\mathit{\sigma }}^{\mathrm{2}}}\right)\mathit{,}$(B.1)In this appendix, we aim to derive the theoretical expression of the moments of the Rician distribution (namely a and σ), in both multimode and singlemode cases. Following Canales & Cagigal (1999), the quantity a and σ can be expressed from the moments of the real (A_r) and imaginary (A_i) parts of the complex instantaneous interferometric Strehl ratio (i.e. such as ): $\begin{array}{ccc}{\mathit{a}}^{\mathrm{4}}& \mathrm{=}& \mathrm{⟨}{\mathit{A}}_{\mathit{r}}{\mathrm{⟩}}_{\mathrm{\Phi }}^{\mathrm{4}}\mathrm{+}\mathrm{2}\mathrm{⟨}{\mathit{A}}_{\mathit{r}}{\mathrm{⟩}}_{\mathrm{\Phi }}^{\mathrm{2}}\mathrm{(}{\mathit{\sigma }}_{\mathit{i}}^{\mathrm{2}}\mathrm{-}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{)}\mathrm{-}\mathrm{(}{\mathit{\sigma }}_{\mathit{i}}^{\mathrm{2}}\mathrm{-}{\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}{\mathrm{)}}^{\mathrm{2}}\\ \mathrm{2}{\mathit{\sigma }}^{\mathrm{2}}& \mathrm{=}& {\mathit{\sigma }}_{\mathit{r}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma }}_{\mathit{i}}^{\mathrm{2}}\mathrm{+}\mathrm{⟨}{\mathit{A}}_{\mathit{r}}{\mathrm{⟩}}_{\mathrm{\Phi }}^{\mathrm{2}}\mathrm{-}{\mathit{a}}^{\mathrm{2}}\end{array}$where and .

B.1. Multimode case

In multimode interferometry we have I =  |T₁₂| ², hence $\begin{array}{ccc}{\mathit{A}}_{\mathit{r}}& \mathrm{=}& \mathrm{Re}\mathrm{\left(}{\mathit{T}}_{\mathrm{12}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}\mathrm{d}{r}}{{\mathrm{\Sigma }}_{\mathit{P}}}\end{array}$(B.4)$\begin{array}{ccc}{\mathit{A}}_{\mathit{i}}& \mathrm{=}& \mathrm{Im}\mathrm{\left(}{\mathit{T}}_{\mathrm{12}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\sin \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}\mathrm{d}{r}}{{\mathrm{\Sigma }}_{\mathit{P}}}\mathrm{·}\end{array}$(B.5)Following the same approach as in appendix A.3, we compute the first and second order moments of these quantities. $\begin{array}{ccc}\mathrm{⟨}{\mathit{A}}_{\mathit{r}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \mathrm{Re}\mathrm{\left(}{\mathit{T}}_{\mathrm{12}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}}{{\mathrm{\Sigma }}_{\mathit{P}}}\mathrm{=}{\exp }^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\end{array}$(B.6)$\begin{array}{ccc}\mathrm{⟨}{\mathit{A}}_{\mathit{i}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \mathrm{Im}\mathrm{\left(}{\mathit{T}}_{\mathrm{12}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{⟨}\sin \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}}{{\mathrm{\Sigma }}_{\mathit{P}}}\mathrm{=}\mathrm{0}\end{array}$(B.7)$\begin{array}{ccc}\mathrm{⟨}{\mathit{A}}_{\mathit{r}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}}\\ & & \mathrm{using} \cos \mathrm{\left(}\mathit{a}\mathrm{\right)}\cos \mathrm{\left(}\mathit{b}\mathrm{\right)}\mathrm{=}\mathrm{\left(}\cos \mathrm{\right(}\mathit{a}\mathrm{-}\mathit{b}\mathrm{)}\mathrm{+}\cos \mathrm{(}\mathit{a}\mathrm{+}\mathit{b}\mathrm{\left)}\mathrm{\right)}\mathit{/}\mathrm{2}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}}\\ & & \mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}}\\ & & \mathrm{changingaxesreference:} {\mathit{P}}_{\mathrm{1}}\mathit{,} {\mathit{P}}_{\mathrm{2}} \mathrm{centeredon}\mathrm{0}\mathit{,}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}^{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\exp }^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\mathrm{\int }\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}^{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\end{array}$(B.8)$\begin{array}{ccc}\mathrm{⟨}{\mathit{A}}_{\mathit{i}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}\sin \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}\sin \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}}\\ & & \mathrm{using} \sin \mathrm{\left(}\mathit{a}\mathrm{\right)}\sin \mathrm{\left(}\mathit{b}\mathrm{\right)}\mathrm{=}\mathrm{\left(}\cos \mathrm{\right(}\mathit{a}\mathrm{-}\mathit{b}\mathrm{)}\mathrm{-}\cos \mathrm{(}\mathit{a}\mathrm{+}\mathit{b}\mathrm{\left)}\mathrm{\right)}\mathit{/}\mathrm{2}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}}\\ & & \mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}}\\ & & \mathrm{changingaxesreference:} {\mathit{P}}_{\mathrm{1}}\mathit{,} {\mathit{P}}_{\mathrm{2}} \mathrm{centeredon}\mathrm{0}\mathit{,}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}^{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\exp }^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\mathrm{\int }\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}\mathrm{[}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{]}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}^{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\end{array}$(B.9)Using Eqs. (B.6, B.7, B.8, B.9) in Eqs. (B.2, B.3), we obtain the theoretical expressions of the parameters a and σ of the Rician distribution is the multimode case, as summarized in Table B.1 of the appendix.

B.2. Singlemode case

In singlemode interferometry we have I =  |ρ₁₂| ², hence $\begin{array}{ccc}{\mathit{A}}_{\mathit{r}}& \mathrm{=}& \mathrm{Re}\mathrm{\left(}{\mathit{\rho }}_{\mathrm{12}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\end{array}$(B.10)$\begin{array}{ccc}{\mathit{A}}_{\mathit{i}}& \mathrm{=}& \mathrm{Im}\mathrm{\left(}{\mathit{\rho }}_{\mathrm{12}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\sin \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathit{.}\end{array}$(B.11)Following the same formalism as above, we compute the first and second order moments of these quantities. $\begin{array}{ccc}\mathrm{⟨}{\mathit{A}}_{\mathit{r}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathrm{=}{\exp }^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\end{array}$(B.12)$\begin{array}{ccc}\mathrm{⟨}{\mathit{A}}_{\mathit{i}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{⟨}\sin \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{f}}{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}}\mathrm{=}\mathrm{0}\end{array}$(B.13)$\begin{array}{ccc}\mathrm{⟨}{\mathit{A}}_{\mathit{r}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{using} \cos \mathrm{\left(}\mathit{a}\mathrm{\right)}\cos \mathrm{\left(}\mathit{b}\mathrm{\right)}\mathrm{=}\mathrm{\left(}\cos \mathrm{\right(}\mathit{a}\mathrm{-}\mathit{b}\mathrm{)}\mathrm{+}\cos \mathrm{(}\mathit{a}\mathrm{+}\mathit{b}\mathrm{\left)}\mathrm{\right)}\mathit{/}\mathrm{2}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{changeofvariables} {r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\to }\mathit{u,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\to }{u}\mathrm{\prime }\mathrm{:}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{+}{\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\end{array}$(B.14)$\begin{array}{ccc}\mathrm{⟨}{\mathit{A}}_{\mathit{i}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}\sin \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\left)}\mathrm{\right)}\sin \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\left)}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{using} \sin \mathrm{\left(}\mathit{a}\mathrm{\right)}\sin \mathrm{\left(}\mathit{b}\mathrm{\right)}\mathrm{=}\mathrm{\left(}\cos \mathrm{\right(}\mathit{a}\mathrm{-}\mathit{b}\mathrm{)}\mathrm{-}\cos \mathrm{(}\mathit{a}\mathrm{+}\mathit{b}\mathrm{\left)}\mathrm{\right)}\mathit{/}\mathrm{2}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}\cos \mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathit{r}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{changeofvariables} {r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\to }{u}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\to }{u}\mathrm{\prime }\mathrm{:}\\ & \mathrm{=}& \frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{-}{\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\mathit{.}\end{array}$(B.15)Putting Eqs. (B.12, B.13, B.14, B.15) in Eqs. (B.2, B.3), we obtain the theoretical expressions of the parameters a and σ of the Rician distribution is the singlemode case, as summarized in Table B.1 of the appendix.

Appendix C: Astrometric phase error

The astrometric phase error writes ${\mathit{\sigma }}_{{\mathrm{\Delta }}_{\mathit{\phi }}^{\mathrm{\Delta }\mathit{\alpha h}}}^{\mathrm{2}}\mathrm{=}\mathrm{2}{\mathit{\sigma }}_{\mathit{\phi }}^{\mathrm{2}}\mathrm{-}\mathrm{2}\mathrm{cov}\mathrm{\left(}{\mathit{\phi }}_{\mathit{s}}\mathit{,}{\mathit{\phi }}_{\mathit{r}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}\mathit{,}$(C.1)where φ_s, and are the partially AO-corrected phase of the astrophysical target and the phase estimated from the off-axis reference source respectively. Δα is the angular separation between both sources, and h is the altitude of the turbulent layer. Since has been computed in previous sections, we now need to compute the covariance part of Eq. (C.1). The formal expression of the covariance is given by Chelli (1989): $\mathrm{cov}\mathrm{\left(}{\mathit{\phi }}_{\mathit{s}}\mathit{,}{\mathit{\phi }}_{\mathit{r}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{Re}\mathrm{\left[}\mathit{E}\mathrm{\right(}\mathit{Q}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}^{\mathrm{\ast }}\mathrm{\left)}\mathrm{\right]}\mathrm{-}\mathrm{Re}\mathrm{\left[}\mathit{E}\mathrm{\right(}\mathit{Q}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\left)}\mathrm{\right]}}{\mathit{E}\mathrm{\left(}\mathit{Q}\mathrm{\right)}\mathit{E}\mathrm{\left(}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}}\mathit{,}$(C.2)where Q_Δαh is the estimator of the coherent flux for the off-axis reference source.

C.1. Multimode case

Using the equations of Sect. A, one can derive the moments of the multimode estimator: $\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathit{Q}\mathrm{\right)}& \mathrm{=}& \mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathit{\phi }}\end{array}$(C.3)$\begin{array}{ccc}\mathit{E}\mathrm{\left(}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}& \mathrm{=}& {\mathit{N}}^{\mathrm{\Delta }\mathit{\alpha h}}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{V}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathit{\phi }}\end{array}$(C.4)$\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathit{Q}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}^{\mathrm{\ast }}\mathrm{\right)}& \mathrm{=}& \mathit{N}{\mathit{N}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}{\mathit{V}}_{\mathrm{12}}\mathrm{(}{\mathit{V}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{)}}^{\mathrm{\ast }}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}\mathrm{(}{f}\mathrm{12}\mathrm{\left)}\mathrm{\right(}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{)}}^{\mathrm{\ast }}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathit{\phi }}\end{array}$(C.5)$\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathit{Q}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}& \mathrm{=}& \mathit{N}{\mathit{N}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}{\mathit{V}}_{\mathrm{12}}{\mathit{V}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathit{\phi }}\end{array}$(C.6)where N^Δαh and are the number of photons (per surface unit and per time unit) and the visibility of the reference source respectively, and with $\begin{array}{ccc}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}{\exp }^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathit{.}\end{array}$(C.7)Since we assume one turbulent layer to be located at the altitude h, we have by definition and , hence: $\begin{array}{ccc}\mathrm{⟨}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}\mathrm{\right(}{r}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{=}{\mathrm{\Sigma }}_{\mathit{P}}{\exp }^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\end{array}$(C.8)Furthermore, we have $\begin{array}{ccc}& & \mathrm{⟨}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}{\mathrm{⟩}}_{\mathrm{\Phi }}\\ & & \mathrm{=}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{=}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}\mathrm{\left|}\mathrm{\right(}{\mathit{\phi }}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{knowingthat} \mathrm{⟨}{\mathit{\phi }}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{\phi }}_{\mathrm{1}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{=}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathit{,} \mathrm{itcomes}\mathit{,}\\ & & \mathrm{=}{\exp }^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{changingaxesreference:} {\mathit{P}}_{\mathrm{1}}\mathit{,} {\mathit{P}}_{\mathrm{2}} \mathrm{centeredon}\mathrm{0}\mathit{,}\\ & & \mathrm{=}{\exp }^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{\int }\left[\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\right]\left[\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\right]\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\end{array}$(C.9)$\begin{array}{ccc}& & \mathrm{⟨}{\mathit{S}}_{\mathrm{12}}\mathrm{\left(}{f}\mathrm{12}\mathrm{\right)}\mathrm{\left(}{\mathit{S}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{)}}^{\mathrm{\ast }}\mathrm{\right(}{f}\mathrm{12}\mathrm{)}{\mathrm{⟩}}_{\mathrm{\Phi }}\\ & & \mathrm{=}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{=}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{⟨}\mathrm{\left|}\mathrm{\right(}{\mathit{\phi }}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}{\mathrm{⟩}}_{\mathrm{\Phi }}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{=}\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{12}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\\ & & \mathrm{changingaxesreference:} {\mathit{P}}_{\mathrm{1}}\mathit{,} {\mathit{P}}_{\mathrm{2}} \mathrm{centeredon}\mathrm{0}\mathit{,}\\ & & \mathrm{=}\mathrm{\int }\left[\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\right]\left[\mathit{P}\mathrm{\left(}{r}\mathit{P}\mathrm{\right(}{{r}}^{\mathrm{\prime }}\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\right]\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathit{.}\end{array}$Putting Eqs. (C.7, C.8, C.9, C.11) into Eqs. (C.5, C.6, C.3, C.4), we finally find that the covariance part of the multimode astrometric phase writes $\mathrm{cov}\mathrm{\left(}{\mathit{\phi }}_{\mathit{s}}\mathit{,}{\mathit{\phi }}_{\mathit{r}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\frac{\mathrm{\int }{\mathrm{\left[}\mathit{P}\mathrm{\right(}{r}\mathrm{\left)}\mathit{P}\mathrm{\right(}{{r}}^{\mathrm{\prime }}\mathrm{\left)}{\mathrm{e}}^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{\right]}}^{\mathrm{2}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{-}{\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{\int }{\mathrm{\left[}\mathit{P}\mathrm{\right(}{r}\mathrm{\left)}\mathit{P}\mathrm{\right(}{{r}}^{\mathrm{\prime }}\mathrm{\left)}{\mathrm{e}}^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{\right]}}^{\mathrm{2}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}^{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}}$(C.12)

C.2. Singlemode case

We once again use formalism of Sect. A to compute the moments of the singlemode estimator: $\begin{array}{ccc}\mathit{E}\mathrm{\left(}\mathit{Q}\mathrm{\right)}& \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}\mathit{N}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}_{\mathrm{0}}{\mathit{V}}_{\mathrm{12}}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}{\mathrm{⟩}}_{\mathit{\phi }}\\ \mathit{E}\mathrm{\left(}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}& \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}{\mathit{N}}^{\mathrm{\Delta }\mathit{\alpha h}}\sqrt{{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}}{\mathit{\rho }}_{\mathrm{0}}{\mathit{V}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{⟩}}_{\mathit{\phi }}\\ \mathit{E}\mathrm{\left(}\mathit{Q}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}^{\mathrm{\ast }}\mathrm{\right)}& \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}\mathit{N}{\mathit{N}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{0}}^{\mathrm{2}}{\mathit{V}}_{\mathrm{12}}\mathrm{(}{\mathit{V}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{)}}^{\mathrm{\ast }}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}\mathrm{(}{\mathit{\rho }}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{)}}^{\mathrm{\ast }}{\mathrm{⟩}}_{\mathit{\phi }}\\ \mathit{E}\mathrm{\left(}\mathit{Q}{\mathit{Q}}_{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}& \mathrm{=}& {\mathrm{\Sigma }}_{\mathit{P}}^{\mathrm{2}}\mathit{N}{\mathit{N}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathit{t}}_{\mathrm{1}}{\mathit{t}}_{\mathrm{2}}{\mathit{\rho }}_{\mathrm{0}}^{\mathrm{2}}{\mathit{V}}_{\mathrm{12}}{\mathit{V}}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}{\mathit{\rho }}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{⟩}}_{\mathit{\phi }}\mathit{,}\end{array}$with $\begin{array}{ccc}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& {\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\\ \mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& {\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathit{.}\end{array}$We also have $\begin{array}{ccc}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}{\mathit{\rho }}_{\mathrm{12}}^{\mathrm{\Delta }\mathit{\alpha h}}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}\mathrm{\right(}{r}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{1}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{{f}}^{\mathrm{\prime }}}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & \mathrm{=}& {\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{changeofvariables} {r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\to }{u}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\to }{u}\mathrm{\prime }\mathrm{:}\\ & \mathrm{=}& {\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\end{array}$(C.19)$\begin{array}{ccc}\mathrm{⟨}{\mathit{\rho }}_{\mathrm{12}}\mathrm{(}{\mathit{\rho }}_{\mathrm{12}}^{\mathbf{\Delta }{\alpha }\mathit{h}}{\mathrm{)}}^{\mathrm{\ast }}{\mathrm{⟩}}_{\mathrm{\Phi }}& \mathrm{=}& \frac{{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}\mathrm{⟨}{\exp }^{\mathit{i}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{1}}\mathrm{\right(}{r}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{1}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{-}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}\mathrm{+}{\mathit{\phi }}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{\left)}\mathrm{\right)}}{\mathrm{⟩}}_{\mathrm{\Phi }}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{{f}}^{\mathrm{\prime }}}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & \mathrm{=}& \frac{\int {\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathrm{)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathrm{+}\mathit{\lambda }{f}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{d}{f}\mathrm{d}{f}\prime }{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\\ & & \mathrm{changeofvariables} {r}\mathrm{+}\mathit{\lambda }{f}\mathrm{\to }{u}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathit{\lambda }{f}\mathrm{\prime }\mathrm{\to }{u}\mathrm{\prime }\mathrm{:}\\ & \mathrm{=}& \frac{\mathrm{\int }{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{r}\mathrm{\prime }}{{{\mathrm{[}}^{\mathrm{\int }}{\mathit{P}}_{\mathrm{1}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{r}\mathrm{\int }{\mathit{P}}_{\mathrm{2}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{2}}}\mathit{.}\end{array}$(C.20)Putting Eqs. (C.17, C.18, C.19, C.20) into Eqs. (C.15, C.16, C.13, C.14), we find that the covariance part of the singlemode astrometric phase writes $\mathrm{cov}\mathrm{\left(}{\mathit{\phi }}_{\mathit{s}}\mathit{,}{\mathit{\phi }}_{\mathit{r}}^{\mathrm{\Delta }\mathit{\alpha h}}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\frac{{{\mathrm{[}}^{\mathrm{\int }}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathrm{e}}^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{{r}\mathrm{\prime }}^{\mathrm{]}}}^{\mathrm{2}}\mathrm{-}{\mathrm{e}}^{\mathrm{-}\mathrm{4}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}{{\mathrm{[}}^{\mathrm{\int }}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathit{P}\mathrm{\left(}{{r}}^{\mathrm{\prime }}\mathrm{\right)}{\mathrm{e}}^{\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{+}\mathbf{\Delta }{\alpha }\mathit{h}\mathrm{)}}\mathrm{d}{r}\mathrm{d}{{r}\mathrm{\prime }}^{\mathrm{]}}}^{\mathrm{2}}}{{{\mathrm{[}}^{\mathrm{\int }}\mathit{P}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{d}{{r}}^{\mathrm{]}}}^{\mathrm{4}}{\mathrm{e}}^{\mathrm{-}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}}\mathit{.}$(C.21)

Appendix D: Phase spatial fluctuations in partial AO correction

The spatial fluctuations of the atmospheric phase can be characterized by the computation of the so-called spatial structure function , defined as the variance of the phase difference taken at two radii r and : ${\mathrm{𝒟}}_{\mathit{\phi }}\mathrm{\left(}{r}\mathit{,}{r}\mathrm{\prime }\mathrm{\right)}\mathrm{=}{\mathrm{𝒟}}_{\mathit{\phi }}\mathrm{(}{r}\mathrm{-}{r}\mathrm{\prime }\mathrm{)}\mathrm{=}\mathrm{⟨}\mathrm{\left|}\mathit{\phi }\mathrm{\right(}{r}\mathrm{)}\mathrm{-}\mathit{\phi }\mathrm{(}{{r}}^{\mathrm{\prime }}\mathrm{)}{\mathrm{|}}^{\mathrm{2}}\mathrm{⟩}\mathit{.}$(D.1)In the fully turbulent case, and assuming Kolmogorov’s description of the atmosphere, the phase structure function can be written (Roddier 1979) ${\mathrm{𝒟}}_{\mathit{\phi }}\mathrm{\left(}{\rho }\mathrm{\right)}\mathrm{=}\mathrm{6.88}{\left(\frac{{\rho }}{{\mathit{r}}_{\mathrm{0}}}\right)}^{\mathrm{5}\mathit{/}\mathrm{3}}\mathit{,}$(D.2)r₀ being the fried parameter, which gives the ultimate spatial resolution achievable with atmospheric turbulence, that is θ_turb = λ/r₀. When partial AO correction is applied, low (spatial) orders of the corrugated wavefront are real-time compensated thanks to a deformable mirror. The variance of the residual turbulent phase φ^r(r) not corrected by the system, can be approximated as following, providing a full correction of the first J ≪ 1 Zernike modes of the turbulence Noll (1976): ${\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}\mathrm{\simeq }\mathrm{0.2944}{\mathit{J}}^{\mathrm{-}\sqrt{\mathrm{3}\mathit{/}\mathrm{2}}}{\left(\frac{\mathit{D}}{{\mathit{r}}_{\mathrm{0}}}\right)}^{\mathrm{5}\mathit{/}\mathrm{3}}\mathrm{·}$(D.3)Furthermore, the phase structure function of the residual phase can be written as ${\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}{\rho }\mathrm{\right)}\mathrm{=}\mathrm{2}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}\left[\mathrm{1}\mathrm{-}\frac{\mathrm{⟨}{\mathit{\phi }}^{\mathit{r}}\mathrm{\left(}{r}\mathrm{\right)}\mathrm{-}{\mathit{\phi }}^{\mathit{r}}\mathrm{(}{r}\mathrm{+}{\rho }\mathrm{)}\mathrm{⟩}}{{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\right]\mathrm{·}$(D.4)For ρ bigger than typically the distance between the mirror’s actuators, the phases are uncorrelated, that is  ⟨ φ^r(r) − φ^r(r + ρ) ⟩  = 0, and consequently saturates at . The long exposure AO corrected transfer function of the atmosphere is by definition $\mathit{B}\mathrm{\left(}{f}\mathrm{\right)}\mathrm{=}\exp \left[\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{{\mathit{\phi }}_{\mathit{r}}}\mathrm{\left(}\mathit{\lambda }{f}\mathrm{\right)}\right]\mathit{.}$(D.5)Taking into account the properties of the phase structure function of the residual phase, B(f) can be decomposed in two parts (Conan 1994): one low-frequency term – namely the halo (FTO_halo) – and one high-frequency term saturating at , namely the coherent energy. From this follows that the long exposure transfer function of the telescope T(f) = T₀(f).B(f), where T₀(f) is the transfer function of a perfect telescope, can be in first approximation modelled by a sum of two components: $\begin{array}{ccc}\mathit{T}\mathrm{\left(}\mathit{f}\mathrm{\right)}& \mathrm{=}& {\mathit{T}}_{\mathrm{0}}\mathrm{\left(}\mathit{f}\mathrm{\right)}\mathit{.}\mathrm{[}{\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{+}\mathit{FT}{\mathit{O}}_{\mathrm{halo}}\mathrm{(}\mathit{f}\mathrm{\left)}\mathrm{\right]}\\ & \mathrm{\simeq }& {\mathit{T}}_{\mathrm{0}}\mathrm{\left(}\mathit{f}\mathrm{\right)}\mathit{.}\mathrm{[}{\mathrm{e}}^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}{\exp }^{\mathrm{-}{\mathit{\sigma }}_{{\mathit{\phi }}_{\mathit{r}}}^{\mathrm{2}}}\mathrm{\left)}{\exp }^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{𝒟}}_{\mathit{\phi }}\mathrm{\left(}\mathit{\lambda }{f}\mathrm{\right)}}\mathrm{\right]}\mathit{.}\end{array}$(D.6)Equation (D.4) also demonstrates that for interferometers equipped with AO systems, the phases and of telescopes 1 and 2 are always uncorrelated. As a consequence, the phase structure function of the residual piston phase is twice the phase structure function of the residual phase over one single telescope (assuming equivalent AO systems for both telescopes), that is . Indeed, even if no AO correction is applied, the phases φ₁ and φ₂ can still be considered to be uncorrelated, as soon as the baseline is longer than the outer scale of the atmosphere ℒ₀. This is for example the case for the VLTI at Paranal, where the outer scale has been estimated to be ℒ₀ ≃ 22 m (Martin et al. 2000).

All Tables

All Figures