Complex visibility ( $V (u, v)$ )

A complex visibility is a calibrated complex correlation of the signals received at a particular interferometric baseline. In a narrow field, with scalar radiation, the far-field approximation for an outgoing source, a narrow frequency channel, negligible noncoplanar phase curvature and a steady sky of mutually incoherent emitters,

V (u, v) = \iint A (l, m) I (l, m) e^{- 2 πi (u l + v m)} d l d m .

(1)

Here

l, m

are small dimensionless sky-direction offsets,

I

is brightness in the chosen channel, and

A

is the primary beam. This is the Fourier transform of beam-weighted brightness. The two-dimensional formula requires those approximations: a wide field generally needs the third baseline coordinate and curved sky geometry.

For an explicit one-dimensional derivation, set the primary beam to one over the small field, and place independent emitters at angular offsets

θ_{j}

. Choose a wave phase convention in which, after removing the phase centre's delay, their received complex amplitudes are

E_{1} (t) = \sum_{j} a_{j} (t), E_{2} (t) = \sum_{j} a_{j} (t) e^{2 πi u θ_{j}}, ⟨ a_{j} a_{k}^{*} ⟩ = I_{j} δ_{jk} .

(2)

Here

δ_{jk}

is the Kronecker delta and angle brackets denote an expected value estimated by time averaging a steady signal. The last equality states mutual incoherence: different emitters have zero averaged cross term, although each emitter's two received copies remain correlated. Multiplying and averaging gives

V (u) = ⟨ E_{1} E_{2}^{*} ⟩ = \sum_{j, k} ⟨ a_{j} a_{k}^{*} ⟩ e^{- 2 πi u θ_{k}} = \sum_{j} I_{j} e^{- 2 πi u θ_{j}} .

(3)

For continuous brightness this becomes

V (u) = \int I (θ) e^{- 2 πi u θ} d θ

. Real brightness implies the Hermitian symmetry

V (- u) = V (u)^{*}

; it does not require brightness to be even. In particular, an asymmetric real source usually has a nonzero imaginary complex visibility. At zero baseline,

V (0) = \sum_{j} I_{j}

is the total beam-weighted source strength. Ordinary separated-element correlations need not measure that zero-baseline value.

Table of contents

Visibility of a one-dimensional uniform source ( $V / V (0) = sinc (π b_{p} Δ α / λ)$ )

 0  0

A centred uniform source of angular width

Δ α

has normalized visibility

V / V (0) = sinc (π b_{p} Δ α / λ)

, with

b_{p}

the projected baseline along that angular coordinate and

sinc x = sin x / x

. The result is the integral of the point-source complex phase over source brightness. Its first null occurs at width

λ /∣ b_{p} ∣

; negative values are phase reversals rather than negative brightness.

Visibility of a uniform rectangular source ( $V = S_{ν} sinc (π Θ_{x} u) sinc (π Θ_{y} v)$ )

 0  0

With constant surface brightness over a centred angular rectangle of widths

Θ_{x}, Θ_{y}

, and a nearly uniform primary beam over that rectangle, the Fourier transform separates into two one-dimensional integrals. The result is

V (u, v) = S_{ν} sinc (π Θ_{x} u) sinc (π Θ_{y} v)

, where

sinc x = sin x / x

. The zero-spacing value is the integrated spectral flux density, while the first axial null is at

∣ u ∣ = 1/ Θ_{x}

. Long baselines therefore do not measure the total flux of a resolved rectangle.

Fringe stopping ( $V_{stop} = V e^{- i ϕ_{ref}}$ )

 0  0

Electronic compensation of the time-dependent reference geometric delay multiplies each complex visibility by an opposite reference phase. A stationary point source at the phase centre then has constant corrected phase despite Earth rotation. antenna pointing alone does not perform this correction. Off-axis sources retain the relative fringe evolution that encodes their angular offsets.

Field-dependent averaging losses in radio interferometry ( $δ ν Bθ / c ≪ 1, Δ t Ω_{E} Bθ / λ ≪ 1$ )

 0  0

After delay tracking and fringe stopping at a phase center, an off-axis direction retains delay of order

Bθ / c

and rotation-induced fringe-rate offset of order

Ω_{E} Bθ / λ

. Averaging across a wide channel or long integration then decorrelates that direction. The displayed inequalities give order-of-magnitude small-smearing conditions; exact coefficients depend on baseline geometry, averaging window and allowed loss. A larger field or longer baseline therefore demands finer frequency and time sampling.

Bandwidth decorrelation of a point source ( $V_{avg} = V (ν_{0}) sinc (π Δ ντ)$ )

 0  0

A point source away from the tracked direction has residual geometric delay

τ

. Averaging its frequency-dependent phase over a top-hat band gives the displayed sinc attenuation times its central-frequency complex phase. The first bandwidth null is complete cancellation, not a small-loss operating limit. Fine frequency channels preserve the information needed for later frequency-aware imaging.

Primary beam ( $A (l, m)$ )

 0  0

The primary beam is the individual antenna pair's directional sensitivity envelope in the complex visibility equation. For identical, identically pointed elements it is the individual element's power pattern. Its angular width sets the useful field of view; the much longer interferometric baselines set finer measured scales. The primary beam and the dirty beam describe different responses.

Complex visibility ( $V (u, v)$ )

Visibility of a one-dimensional uniform source ( $V / V (0) = sinc (π b_{p} Δ α / λ)$ )

Visibility of a uniform rectangular source ( $V = S_{ν} sinc (π Θ_{x} u) sinc (π Θ_{y} v)$ )

Fringe stopping ( $V_{stop} = V e^{- i ϕ_{ref}}$ )

Field-dependent averaging losses in radio interferometry ( $δ ν Bθ / c ≪ 1, Δ t Ω_{E} Bθ / λ ≪ 1$ )

Bandwidth decorrelation of a point source ( $V_{avg} = V (ν_{0}) sinc (π Δ ντ)$ )

Primary beam ( $A (l, m)$ )

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