Fiber optic interferometry: statistics of visibility and closure phase

E. Tatulli; A. Chelli

doi:10.1364/JOSAA.22.001589

Journal of the Optical Society of America A
Vol. 22,
Issue 8,
pp. 1589-1599
(2005)
•https://doi.org/10.1364/JOSAA.22.001589

Fiber optic interferometry: statistics of visibility and closure phase

E. Tatulli and A. Chelli

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E. Tatulli and A. Chelli, "Fiber optic interferometry: statistics of visibility and closure phase," J. Opt. Soc. Am. A 22, 1589-1599 (2005)

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Abstract

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 $1 ∕ 2$ for the visibilities and $1 ∕ 3$ 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 $1 ∕ 2$ 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.

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Observables	Variance of Observables (Point Source)
	Photon Noise Regime $(\bar{K} ⪢ 1)$		Detector Noise Regime $(\bar{K} ⪡ 1)$
	Full AO Correction	No AO Correction	Detector Noise Regime $(\bar{K} ⪡ 1)$
$\tilde{V^{2}}$	$[\frac{2 N_{tel}}{(1 - τ)} + \frac{2}{τ}] \frac{N_{tel}}{ρ_{0} \bar{S K}}$	$[\frac{2 N_{tel} + 4}{(1 - τ)} + \frac{4}{τ}] \frac{N_{tel}}{ρ_{0} \bar{S K}}$	$[\frac{3 N_{pix} σ^{4} + N_{pix}^{2} σ^{4}}{{(1 - τ)}^{4}}] \frac{N_{tel}^{4}}{ρ_{0}^{4} {\bar{S}}^{4} {\bar{K}}^{4}}$
${\tilde{ϕ}}_{B}$	$[\frac{3 N_{tel} - 6}{2}] \frac{N_{tel}}{ρ_{0} \bar{S K}}$	$[\frac{3 N_{tel} - 3}{2}] \frac{N_{tel}}{ρ_{0} \bar{S K}}$	$[\frac{9 N_{pix} σ^{4} + 3 N_{pix}^{2} σ^{4}}{2}] \frac{N_{tel}^{4}}{ρ_{0}^{4} {\bar{S}}^{4} {\bar{K}}^{4}}$

Observables	Correlation Coefficient ρ (point source)
	Photon Noise Regime $(\bar{K} ⪢ 1)$		Detector Noise Regime $(\bar{K} ⪡ 1)$
	Full AO Correction	No AO Correction	Detector Noise Regime $(\bar{K} ⪡ 1)$
$\tilde{V^{2}}$
If closure	$⩽ 1 / 2$	$⩽ 1 / 4$	$\frac{3}{3 + N_{pix}}$
Otherwise	0	0	$\frac{3}{3 + N_{pix}}$
${\tilde{ϕ}}_{B}$
If baseline in common	$1 / 3$	$⩽ 1 / 6$	$\frac{3}{9 + N_{pix}}$
Otherwise	0	0	0

Symbol	Telescopes	Average Projected Baseline
+	UT2–UT3	$45 m$
×	UT1–UT2	$56 m$
◇	UT2–UT4	$83 m$
Δ	UT1–UT3	$100 m$
◻	UT1–UT4	$124 m$

Terms	Covariance Coefficients
Diagonal	$[\frac{4 σ_{S}^{2} + 2 N_{tel} {\bar{S}}^{2}}{(1 - τ) V_{⋆}^{2} (f_{i j}) {\bar{S}}^{3}} + \frac{2 (σ_{S}^{2} + {\bar{S}}^{2})}{τ {\bar{S}}^{3}}] \frac{N_{tel}}{ρ_{0} \bar{K}} + [\frac{2 N_{tel} σ_{S}^{2} + N_{tel}^{2} {\bar{S}}^{2}}{{(1 - τ)}^{2} V_{⋆}^{4} (f_{i j}) {\bar{S}}^{4}} + \frac{4}{{(1 - τ)}^{2} V_{⋆}^{2} (f_{i j}) {\bar{S}}^{2}} + \frac{1}{τ^{2} {\bar{S}}^{2}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}} + \frac{1}{{(1 - τ)}^{3} V_{⋆}^{4} (f_{i j}) {\bar{S}}^{3}} \frac{N_{tel}^{4}}{ρ_{0}^{3} {\bar{K}}^{3}}$
	$+ [\frac{2 N σ^{2}}{{(1 - τ)}^{2} V_{⋆}^{2} (f_{i j}) {\bar{S}}^{2}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}} + [\frac{2 N σ^{2}}{{(1 - τ)}^{3} V_{⋆}^{4} (f_{i j}) {\bar{S}}^{3}}] \frac{N_{tel}^{4}}{ρ_{0}^{3} {\bar{K}}^{3}} + [\frac{3 N σ^{4} + N^{2} σ^{4}}{{(1 - τ)}^{4} V_{⋆}^{4} (f_{i j}) {\bar{S}}^{4}}] \frac{N_{tel}^{4}}{ρ_{0}^{4} {\bar{K}}^{4}}$
Nondiagonal	$[\frac{2}{{(1 - τ)}^{2} {\bar{S}}^{2}} [\frac{1}{V_{⋆}^{2} (f_{i j})} + \frac{1}{V_{⋆}^{2} (f_{k l})}] + \frac{N_{tel} σ_{S}^{2}}{{(1 - τ)}^{2} V_{⋆}^{2} (f_{i j}) V_{⋆}^{2} (f_{k l}) {\bar{S}}^{4}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}} + [\frac{1}{{(1 - τ)}^{3} V_{⋆}^{2} (f_{i j}) V_{⋆}^{2} (f_{k l}) {\bar{S}}^{3}}] \frac{N_{tel}^{4}}{ρ_{0}^{3} {\bar{K}}^{3}} + [\frac{3 N σ^{4}}{{(1 - τ)}^{4} V_{⋆}^{2} (f_{i j}) V_{⋆}^{2} (f_{k l}) S^{4}}]$
	if: $j = k + \frac{1}{τ \bar{S}} \frac{N_{tel}}{ρ_{0} \bar{K}} + [\frac{2 Re [V_{⋆} (f_{i j}) V_{⋆} (f_{k l}) V_{⋆}^{*} (f_{i l})]}{(1 - τ) V_{⋆}^{2} (f_{i j}) V_{⋆}^{2} (f_{k l}) \bar{S}}] \frac{N_{tel}}{ρ_{0} \bar{K}} + [\frac{V_{⋆}^{2} (f_{i l})}{{(1 - τ)}^{2} V_{⋆}^{2} (f_{i j}) V_{⋆}^{2} (f_{k l}) S^{2}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}}$

Terms	Covariance Coefficients
Diagonal	$[\frac{[N_{tel}^{3} - 2 V_{⋆}^{} (f_{i j}) V_{⋆}^{} (f_{j k}) V_{⋆} (f_{i k})] {\bar{S}}^{3} + 3 N_{tel}^{2} σ_{S}^{2} \bar{S}}{2 {∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{i k}) ∣}^{2} {\bar{S}}^{6}}] \frac{N_{tel}^{3}}{ρ_{0}^{3} {\bar{K}}^{3}} + [\frac{[{∣ V_{⋆} (f_{i j}) ∣}^{2} + {∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i k}) ∣}^{2}] [N_{tel}^{2} {\bar{S}}^{4} + N_{tel} κ_{S}^{2} {\bar{S}}^{2} + 2 σ_{S}^{4}]}{2 {∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{i k}) ∣}^{2} {\bar{S}}^{6}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}}$
	$- [\frac{[{∣ V_{⋆} (f_{i j}) ∣}^{4} + {∣ V_{⋆} (f_{j k}) ∣}^{4} + {∣ V_{⋆} (f_{i k}) ∣}^{4}] [{\bar{S}}^{4} + 2 σ_{S}^{2} {\bar{S}}^{2} + σ_{S}^{4}] + 2 [{∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i k}) ∣}^{2} ({∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i j}) ∣}^{2})] [{\bar{S}}^{4} + σ_{S}^{2} {\bar{S}}^{2}]}{2 {∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{i k}) ∣}^{2} {\bar{S}}^{6}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}}$
	$+ [\frac{[{∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i k}) ∣}^{2} ({∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i j}) ∣}^{2})] [N_{tel} {\bar{S}}^{5} + (N_{tel} + 4) σ_{S}^{2} {\bar{S}}^{3} + 2 σ_{S}^{4} \bar{S}]}{2 {∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{i k}) ∣}^{2} {\bar{S}}^{6}}] \frac{N_{tel}}{ρ_{0} \bar{K}}$
	$- [\frac{2 [V_{⋆} (f_{i j}) V_{⋆} (f_{j k}) V_{⋆}^{*} (f_{i k})] [{∣ V_{⋆} (f_{i j}) ∣}^{2} + {∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i k}) ∣}^{2}] [{\bar{S}}^{5} + 2 σ_{S}^{2} {\bar{S}}^{3} + σ_{S}^{4} S]}{2 {∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{i k}) ∣}^{2} {\bar{S}}^{6}}] \frac{N_{tel}}{ρ_{0} \bar{K}}$
Nondiagonal	$0 if f_{i j} \neq f_{l m}$
	$[\frac{{∣ V_{⋆} (f_{k n}) ∣}^{2} [N_{tel} {\bar{S}}^{3} + 2 σ_{S}^{2} \bar{S}]}{2 V_{⋆} (f_{i j}) V_{⋆} (f_{j k}) ∣ V_{⋆}^{} (f_{i k}) V_{⋆}^{} (f_{l m}) V_{⋆}^{*} (f_{m n}) ∣ V_{⋆} (f_{l n}) {\bar{S}}^{6}}] \frac{N_{tel}^{3}}{ρ_{0}^{3} {\bar{K}}^{3}}$
	$+ [\frac{[{∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{k n}) ∣}^{2}] {\bar{S}}^{4} + [V_{⋆} (f_{k n}) V_{⋆}^{} (f_{m n}) V_{⋆}^{} (f_{m k}) + V_{⋆} (f_{k n}) V_{⋆}^{} (f_{l k}) V_{⋆}^{} (f_{l n})] [N_{tel} {\bar{S}}^{4} + 3 σ_{S}^{2} {\bar{S}}^{2}]}{2 V_{⋆} (f_{i j}) V_{⋆} (f_{j k}) ∣ V_{⋆}^{} (f_{i k}) V_{⋆}^{} (f_{l m}) V_{⋆}^{*} (f_{m n}) ∣ V_{⋆} (f_{l n}) {\bar{S}}^{6}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}}$
	$- [\frac{[{∣ V_{⋆} (f_{i k}) ∣}^{2} {∣ V_{⋆} (f_{i n}) ∣}^{2} + {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{j n}) ∣}^{2}] [{\bar{S}}^{4} + σ_{S}^{2} {\bar{S}}^{2}] + [{∣ V_{⋆} (f_{i k}) ∣}^{2} {∣ V_{⋆} (f_{j n}) ∣}^{2} + {∣ V_{⋆} (f_{i n}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2}]}{2 V_{⋆} (f_{i j}) V_{⋆} (f_{j k}) ∣ V_{⋆}^{} (f_{i k}) V_{⋆}^{} (f_{l m}) V_{⋆}^{*} (f_{m n}) ∣ V_{⋆} (f_{l n}) {\bar{S}}^{6}}] {\bar{S}}^{4} \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}}$
	$[\frac{[V_{⋆} (f_{j k}) V_{⋆}^{} (f_{i k}) V_{⋆}^{} (f_{j n}) V_{⋆} (f_{i n})] [N_{tel} {\bar{S}}^{5} + 4 σ_{S}^{2} {\bar{S}}^{3}] + [V_{⋆} (F_{k n}) V_{⋆}^{} (F_{i n}) V_{⋆}^{} (F_{i k}) + V_{⋆} (F_{k n}) V_{⋆}^{} (F_{j k}) V_{⋆}^{} (F_{j n})] {∣ V_{⋆} (f_{i j}) ∣}^{2} [{\bar{S}}^{5} + σ_{S}^{2} {\bar{S}}^{3}]}{2 V_{⋆} (f_{i j}) V_{⋆} (f_{j k}) ∣ V_{⋆}^{} (f_{i k}) V_{⋆}^{} (f_{l m}) V_{⋆}^{*} (f_{m n}) ∣ V_{⋆} (f_{l n}) {\bar{S}}^{6}}] \frac{N_{tel}}{ρ_{0} \bar{K}}$
	$- [\frac{[V_{⋆} (f_{i j}) V_{⋆} (f_{j k}) V_{⋆}^{} (f_{i k}) ({∣ V_{⋆} (f_{i n}) ∣}^{2} + {∣ V_{⋆} (f_{j n}) ∣}^{2}) + V_{⋆} (f_{i j}) V_{⋆} (f_{j n}) V_{⋆}^{} (f_{i n}) ({∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i k}) ∣}^{2})] [{\bar{S}}^{5} + σ_{S}^{2} {\bar{S}}^{3}]}{2 V_{⋆} (f_{i j}) V_{⋆} (f_{j k}) ∣ V_{⋆}^{} (f_{i k}) V_{⋆}^{} (f_{l m}) V_{⋆}^{*} (f_{m n}) ∣ V_{⋆} (f_{l n}) {\bar{S}}^{6}}] \frac{N_{tel}}{ρ_{0} \bar{K}} if f_{i j} = f_{l m}$

Terms	Covariance Coefficients
Diagonal	$[\frac{[{∣ V_{⋆} (f_{i j}) ∣}^{2} + {∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i k}) ∣}^{2}] [3 N σ^{4} + N^{2} σ^{4}] {\bar{S}}^{2}}{2 {∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{i k}) ∣}^{2} {\bar{S}}^{6}}] \frac{N_{tel}^{4}}{ρ_{0}^{4} {\bar{K}}^{4}}$
	$+ [\frac{[{∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{i j}) ∣}^{2} + {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} + {∣ V_{⋆} (f_{i k}) ∣}^{2} {∣ V_{⋆} (f_{i j}) ∣}^{2}] N σ^{2} [{\bar{S}}^{4} + σ_{S}^{2} {\bar{S}}^{2}]}{2 {∣ V_{⋆} (f_{i j}) ∣}^{2} {∣ V_{⋆} (f_{j k}) ∣}^{2} {∣ V_{⋆} (f_{i k}) ∣}^{2} {\bar{S}}^{6}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}} + N^{3} σ^{6} + 3 N^{2} σ^{6}$
Nondiagonal	$0 if f_{i j} \neq f_{l m}$
	$[\frac{[V_{⋆} (f_{j k}) V_{⋆}^{} (f_{i k}) V_{⋆}^{} (f_{j n}) V_{⋆} (f_{i n})] N σ^{2} {\bar{S}}^{4}}{2 V_{⋆} (f_{i j}) V_{⋆} (f_{j k}) ∣ V_{⋆}^{} (f_{i k}) V_{⋆}^{} (f_{l m}) V_{⋆}^{*} (f_{m n}) ∣ V_{⋆} (f_{l n}) {\bar{S}}^{6}}] \frac{N_{tel}^{2}}{ρ_{0}^{2} {\bar{K}}^{2}} + 3 N^{2} σ^{6} if f_{i j} = f_{l m}$

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Equations (53)

Journal of the Optical Society of America A