Interferometric observations with three telescopes or more provide two observables: closure phase information and visibility measurements. When single-mode interferometers are used, both observables have to be redefined in the light of the coupling phenomenon between the incoming wave front and the fiber. We introduce the estimator of both the so-called modal visibility and the modal closure phase. Then we compute the statistics of the two observables in the presence of partial correction by adaptive optics, paying attention to the correlation between the measurements. We find that the correlation coefficients are mostly zero and in any case are never greater than for the visibilities and for the closure phases. From this theoretical analysis, a data-reduction process using classic least-squares minimization is investigated. In the framework of the AMBER instrument, the three-beam recombiner of the Very Large Telescope Interferometer (VLTI), we simulate the observation of a single Gaussian source and study the performances of the interferometer in terms of diameter measurements. We show that the observation is optimized, i.e., that the signal-to-noise ratio (SNR) of the diameter is maximal when the FWHM of the source is roughly of the mean resolution of the interferometer. We finally point out that, in the case of an observation with three telescopes, neglecting the correlation between the measurements leads to overestimating the SNR by a factor of 2. We infer that in any case this value is an upper limit.
Damien Ceus, Alessandro Tonello, Ludovic Grossard, Laurent Delage, François Reynaud, Harald Herrmann, and Wolfgang Sohler Opt. Express 19(9) 8616-8624 (2011)
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Variance of Visibility and Closure Phase for a Point Sourcea
Observables
Variance of Observables (Point Source)
Photon Noise Regime
Detector Noise Regime
Full AO Correction
No AO Correction
corresponds to the number of pixels that sample the interferogram. is required by the Shannon criterion. σ is the detector noise per pixel.
Table 2
Correlation Coefficients of Visibilities and Closure Phasesa
Observables
Correlation Coefficient ρ (point source)
Photon Noise Regime
Detector Noise Regime
Full AO Correction
No AO Correction
If closure
Otherwise
0
0
If baseline in common
Otherwise
0
0
0
For the visibility, two cases are considered: one telescope is common to both baselines—hence the triplet of telescopes is forming a triangle (a so-called closure)—or not (see Fig. 2). For the closure phases, two cases are investigated as well: one baseline belongs to both closure phases or not (see Fig. 3).
Table 3
Description of the Considered Telescopes Configurations (Fig. 5)a
Symbol
Telescopes
Average Projected Baseline
+
UT2–UT3
×
UT1–UT2
◇
UT2–UT4
Δ
UT1–UT3
◻
UT1–UT4
The telescopes are the four unit telescopes of the VLTI. The declination of the source is arbitrarily set to . The source is assumed to be observed between and from the zenith.
Table 4
Elements of the Visibility Covariance Matrix [Eq. (A4)]
Terms
Covariance Coefficients
Diagonal
Nondiagonal
if:
Table 5
Elements of the Closure Phase Covariance Matrix in the Photon Noise Regime [Eq. (A6)]
Terms
Covariance Coefficients
Diagonal
Nondiagonal
Table 6
Elements of the Closure Phase Covariance Matrix, in the Detector Noise Regime [Eq. (A6)]
Terms
Covariance Coefficients
Diagonal
Nondiagonal
Tables (6)
Table 1
Variance of Visibility and Closure Phase for a Point Sourcea
Observables
Variance of Observables (Point Source)
Photon Noise Regime
Detector Noise Regime
Full AO Correction
No AO Correction
corresponds to the number of pixels that sample the interferogram. is required by the Shannon criterion. σ is the detector noise per pixel.
Table 2
Correlation Coefficients of Visibilities and Closure Phasesa
Observables
Correlation Coefficient ρ (point source)
Photon Noise Regime
Detector Noise Regime
Full AO Correction
No AO Correction
If closure
Otherwise
0
0
If baseline in common
Otherwise
0
0
0
For the visibility, two cases are considered: one telescope is common to both baselines—hence the triplet of telescopes is forming a triangle (a so-called closure)—or not (see Fig. 2). For the closure phases, two cases are investigated as well: one baseline belongs to both closure phases or not (see Fig. 3).
Table 3
Description of the Considered Telescopes Configurations (Fig. 5)a
Symbol
Telescopes
Average Projected Baseline
+
UT2–UT3
×
UT1–UT2
◇
UT2–UT4
Δ
UT1–UT3
◻
UT1–UT4
The telescopes are the four unit telescopes of the VLTI. The declination of the source is arbitrarily set to . The source is assumed to be observed between and from the zenith.
Table 4
Elements of the Visibility Covariance Matrix [Eq. (A4)]
Terms
Covariance Coefficients
Diagonal
Nondiagonal
if:
Table 5
Elements of the Closure Phase Covariance Matrix in the Photon Noise Regime [Eq. (A6)]
Terms
Covariance Coefficients
Diagonal
Nondiagonal
Table 6
Elements of the Closure Phase Covariance Matrix, in the Detector Noise Regime [Eq. (A6)]