Radio interferometry

Radio interferometry measures the spatial coherence of received radio waves. At two antennas, a receiver preserves the relative wave phase, compensates the geometric delay for a chosen phase centre, and averages a product of one signal with the complex conjugate of the other. In a complex-signal description this correlation is $⟨E_1{E}_2^{\ast }⟩$. Its real and imaginary parts carry two complementary components of the complex visibility; measuring only the received powers discards the relative wave phase needed for general imaging.

A finite frequency band makes delay compensation essential. If the residual delay is $\Delta \tau$, the residual wave phase varies by $2\pi \Delta \nu \Delta \tau$ across a band of width $\Delta \nu$. Averaging these differently oriented complex contributions reduces the correlation. For an ideal rectangular band the multiplicative factor is $\sin (\pi \Delta \nu \Delta \tau )/(\pi \Delta \nu \Delta \tau )$. A phase correction at just the central frequency cannot remove this variation across the band. Long time averages can likewise smear a changing signal.

A radio interferometer combines voltage signals from separated antennas to measure the spatial coherence of radio waves. A correlator compares signals after correcting for the geometrical delay associated with an observing direction. The resulting complex visibility retains relative wave phase information that independent power measurements lose.

For a baseline vector $\mathbf{b}$ and source direction $\mathbf{s}$, the geometrical delay is $\tau =\mathbf{b}\cdot \mathbf{s}/c$. The angular resolution scale is roughly $\lambda /B$, where $B$ is the longest relevant baseline. Multiple baselines and the changing orientation supplied by Earth's rotation sample different spatial frequencies for aperture synthesis; a single pair does not supply a complete image.

The Very Long Baseline Array is a radio-telescope array used for very-long-baseline interferometry. In a solar-deflection experiment, differential astrometry of nearby compact sources compares their apparent directions as their angular separation from the Sun changes. Measurements at several frequencies can distinguish dispersive refraction in the solar corona from achromatic gravitational deflection.^[ref]

very-long-baseline interferometry correlates accurately time-referenced radio signals recorded at widely separated collectors, giving angular resolution of order wavelength divided by projected baseline. Geometric delays, frequency standards, atmospheric propagation and source structure enter the calibration. Compact maser positions and motions are one application.

A residual delay $\delta \tau$ in a rectangular band of width $\Delta \nu$ attenuates the correlated amplitude by $\mathrm{sinc}(\pi \Delta \nu \delta \tau )$. Coarse delay quantization must keep this factor near one; the relevant path scale is $c/\Delta \nu$. Separately, an untracked carrier phase has error $2\pi \delta \ell /\lambda$. Fine phase rotation can remove that central-frequency phase while leaving the band-dependent decorrelation, so delay-step specifications must state both bandwidth and compensation architecture.

Independent visibility samples are combined with weights proportional to inverse noise variance. This maximizes the point-source sensitivity for correctly calibrated samples. Dense regions of the sampled Fourier plane receive more total weight; tapering the long baselines can further emphasize extended structure at the cost of angular resolution. A source-template filter instead also weights by predicted source response.

For $N$ equal antennas, weak-source independent receiver noise, natural weighting and total on-source time $t$, the point-source rms in one polarization of waves is the displayed value. There are $N(N-1)/2$ independent cross-correlation baselines, each with real-component error $\mathrm{SEFD}/\sqrt{2\Delta \nu t}$. Combining $n_{\mathrm{pol}}$ independent polarization of waves estimates reduces the rms by $\sqrt{n_{\mathrm{pol}}}$. Correlation losses divide the sensitivity by an efficiency factor; weighting, flagging and an extended source change the effective sensitivity.^[ref]

The phase centre is the reference sky direction for removing geometric delay and its associated wave phase. An unresolved source exactly there ideally has constant complex visibility after calibration; a displaced source has a baseline-dependent wave phase.

An interferometric baseline is the displacement $\mathbf{b}={\mathbf{r}}_2-{\mathbf{r}}_1$ between two antennas. For a unit line of sight ${\mathbf{s}}_0$ toward the phase centre, its projection onto the plane of the sky isHere $c$ is the speed of light. The signed delay depends on which receiver is delayed; the geometric path difference's magnitude does not. If ${\mathbf{e}}_{\theta }$ is a unit tangent in the direction being imaged, the dimensionless coordinate $u=\mathbf{b}\cdot {\mathbf{e}}_{\theta }/\lambda$ is a spatial frequency, measured in cycles per radian of small sky offset. A single pair gives fringes of angular period $1/|u|$ along that tangent. This period specifies one measured oscillation; it does not specify a whole image's angular resolution or eliminate ambiguous source arrangements.

The left panel separates the component producing the reference delay from the component measuring angular structure. The right panel shows a possible Earth-rotation aperture synthesis track, with its opposite complex conjugate track. Those opposite samples provide the symmetry required by a real brightness distribution; they are not independent extra observations.

The shortest useful projected baseline sets the lowest sampled spatial frequency, of order $b_{\min }/\lambda$, and hence an angular scale $\lambda /b_{\min }$. Operational maximum recoverable source sizes also depend on source shape, coverage, weighting and primary beam. The longest baseline sets a different scale, the synthesized angular resolution. The number of antennas alone does not fix either limiting baseline or supply zero-spacing total power.

The difference in plane-wave propagation time between two antennas is the projected baseline along the source direction divided by the speed of light. Its sign depends on the ordering of the receivers. A monochromatic signal acquires relative phase $2\pi \nu {\tau }_g$; its change with Earth rotation is compensated by fringe stopping for the chosen reference direction.

For an east-west baseline $b$ and phase-centre zenith angle $\theta$ in its vertical plane, path compensation subtracts $b\sin \theta$. An offset $\alpha$ leaves $\delta \ell =b[\sin (\theta +\alpha )-\sin \theta ]\simeq b\alpha \cos \theta$. The complex correlation therefore has phase $2\pi b\alpha \cos \theta /\lambda$, up to receiver-order convention. Equal cable lengths do not remove this sky-dependent path difference.

Earth-rotation aperture synthesis uses Earth's changing orientation to vary a fixed ground baseline's projection onto the plane of the sky. In two dimensions, $u$ and $v$ are the components of ${\mathbf{b}}_{\perp }/\lambda$ along two perpendicular sky tangents. Tracking one field traces a curve in this $(u,v)$ plane; changing the physical baseline supplies other curves. A track samples Fourier transform coordinates, rather than the brightness along a track across the sky.

For a horizontal east–west baseline of length $B$, choose the sky-axis signs so thatwhere $H$ is hour angle and $\delta$ is declination. The ellipse follows by projecting the rotating baseline onto sky tangents. At a celestial pole it is a circle; near the celestial equator its minor axis collapses. Thus the same ground layout constrains different sky directions unequally. Rotation improves coverage, but does not fill every spatial frequency, and synthesis across many hours assumes that the source changes negligibly during that interval.

Radio interferometer calibration estimates instrumental and propagation effects that alter a complex visibility. A simple narrow-field model iswhere $g_1,g_2$ are complex receiver gains and $n_{12}$ represents additive noise. A point-like astronomical source of known position and stable strength constrains these gains: its expected complex visibility has a known wave phase and amplitude. A resolved or variable reference source requires a better source model. Errors in the reference model can be transferred into the inferred sky.

In a proposed extension, Martin Ryle described simultaneous observation of a nearby reference point source to correct atmospheric wave phase errors, using an additional dish at each element. He presented this as a possible extension, rather than the standard One-Mile Telescope arrangement.^[ref]

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,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 ${\theta }_j$. Choose a wave phase convention in which, after removing the phase centre's delay, their received complex amplitudes areHere ${\delta }_{jk}$ 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 givesFor continuous brightness this becomes $V(u)=\int I(\theta )e^{-2\pi iu\theta }d\theta$. Real brightness implies the Hermitian symmetry $V(-u)=V(u{)}^{\ast }$; 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.

A centred uniform source of angular width $\Delta \alpha$ has normalized visibility $V/V(0)=\mathrm{sinc}(\pi b_p\Delta \alpha /\lambda )$, with $b_p$ the projected baseline along that angular coordinate and $\mathrm{sinc}x=\sin x/x$. The result is the integral of the point-source complex phase over source brightness. Its first null occurs at width $\lambda /|b_p|$; negative values are phase reversals rather than negative brightness.

With constant surface brightness over a centred angular rectangle of widths ${\Theta }_x,{\Theta }_y$, and a nearly uniform primary beam over that rectangle, the Fourier transform separates into two one-dimensional integrals. The result is $V(u,v)=S_{\nu }\mathrm{sinc}(\pi {\Theta }_{x}u)\mathrm{sinc}(\pi {\Theta }_{y}v)$, where $\mathrm{sinc}x=\sin x/x$. The zero-spacing value is the integrated spectral flux density, while the first axial null is at $|u|=1/{\Theta }_x$. Long baselines therefore do not measure the total flux of a resolved rectangle.

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.

After delay tracking and fringe stopping at a phase center, an off-axis direction retains delay of order $B\theta /c$ and rotation-induced fringe-rate offset of order ${\Omega }_{E}B\theta /\lambda$. 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.

A point source away from the tracked direction has residual geometric delay $\tau$. 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.

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.