# Interferometers II

Interferometers in Three Dimensions

The VLA (Very Large Array) is a two-dimensional array whose baselines are not confined to an east-west line, but it is nearly coplanar, so "snapshot" observations much shorter than a sidereal day can be treated as two dimensional.  On longer time scales, Earth rotation causes the VLA baselines to fill a three-dimensional volume.

The VLA (Very Large Array) is a Y-shaped and nearly coplanar array of 27 25-m telescopes on the high Plains of San Augustin in New Mexico. At any instant it is a nearly two-dimensional interferometer, so it can make "snapshot" images.  The north-south baselines allow imaging with a nearly circular synthesized beam even near the celestial equator. This photograph shows the "D" configuration spanning about 1 km.  The telescopes can be moved along railroad tracks to form the "C", "B", and "A" configurations spanning 3.4, 11, and 36 km, respectively for higher angular resolution. The VLA is undergoing a major upgrade to become the EVLA (the "E" stands for "Expanded"), with new receivers and a far more powerful and versatile correlator.  The EVLA will be up to an order of magnitude more sensitive than the VLA.  Image credit

The instantaneous beam of the VLA (Very Large Array) has stronger and more extended sidelobes than the beam of a filled aperture.  Note the six diffraction spikes caused by the three straight arms of the VLA.  These sidelobes can be reduced significantly by Earth-rotation aperture synthesis.

The $(u,v,w)$ coordinate system used to describe any baseline vector $\vec{b}$ in three dimensions is shown below.  The $w$ axis is in the reference direction $\hat{s}_0$ usually chosen to contain the source.  The $u$ and $v$ axes point east and north in the $(u,v)$ plane normal to the $w$ axis.  $u$, $v$, and $w$ are the components of $\vec{b}/\lambda$, the baseline vector in wavelength units. An arbitrary unit vector $\hat{s}$ has components $(l,m,n)$ as drawn, where $n = \cos(\theta) = (1 - l^2 -m^2)^{1/2}$. The components $(l,m,n)$ are called direction cosines.

The (u,v,w) coordinate system for interferometers.  The w axis points in the reference direction $\hat{s}_0$ usually containing the source to be imaged.  Projected onto the plane normal to the w axis, $u$ is the east-west baseline in wavelengths and $v$ is the north-south baseline in wavelengths.  $l$, $m$, and $n$ are projections of the unit vector $\hat{s}$ onto the $u$, $v$, and $w$ axes, respectively.

Since $$d \Omega = {dl dm \over (1 -l^2 -m^2)^{1/2}}~,$$ the three-dimensional generalization of Equation 3F1 is $$\bbox[border:3px blue solid,7pt]{V_\nu (u,v,w) = \int \int {I_\nu (l,m) \over (1 -l^2 -m^2)^{1/2}} \exp[-i 2 \pi(ul+vm+wn)] dl dm}\rlap{\quad \rm {(3G1)}}$$ This is not a three-dimensional Fourier transform.

However, if $w = 0$, Equation 3G1 becomes a two-dimensional Fourier transform, which can be inverted to give the source brightness distribution in terms of the measured visibilities: $$\bbox[border:3px blue solid,7pt]{{I_\nu(l,m) \over (1 - l^2 -m^2)^{1/2}} = \int\int V_\nu(u,v,0) \exp[+i 2 \pi (ul +vm)] du dv}\rlap{\quad \rm {(3G2)}}$$ That is the case for a Earth-rotation aperture synthesis by an east-west interferometer if we choose $\hat{s}_0$ to coincide with the Earth's rotation axis, in which case $(1 -l^2-m^2)^{1/2} = \cos\theta = \sec\delta$, where $\delta$ is the declination of the reference position.

For any interferometer, if we consider only directions close to $\hat{s}_0$, then $n = \cos\theta \approx 1 -\theta^2/2$ and
$$V_\nu (u,v,w) \approx \exp(-i 2 \pi w) \int\int {I_\nu (l,m) \over (1 - l^2 -m^2)^{1/2}} \exp[-i 2 \pi (ul +vm - w\theta^2/2)] dl dm~.$$
We can make the factor $\exp(-i 2 \pi w \theta^2/2)$ irrelevant (close to unity) by requiring that $w \theta^2 \ll 1$; that is, by imaging only a small field of view whose radius is $\theta \ll w^{-1/2} \approx (\lambda/b)^{1/2}$.  For example, $\theta \ll 0.01$ radians is sufficient for an interferometer baseline $10^4$ wavelengths long.  Then
$$V_\nu \exp(i 2 \pi w) = \int \int {I_\nu (l, m) \over (1 - l^2 -m^2)^{1/2}} \exp[-i 2 \pi (ul + vm)] dl dm$$ A field wider than $\theta \ll w^{-1/2}$ can be imaged with two-dimensional Fourier transforms by breaking it up into smaller facets, much like a fly's eye, and merging the facets to make the final image.

Sensitivity

The point-source sensitivity of a two-element interferometer can be derived from the radiometer equation for a total-power receiver on a single antenna because a square-law detector is equivalent to a correlator multiplying two identical input voltages supplied by one antenna.  Consider an interferometer with two identical elements, each of which also has a square-law detector, observing a point source.  The correlator multiplies the voltages from the two antennas, while each square-law detector multiplies the voltage from one antenna by itself, so the correlated/detected output voltages of the interferometer and each single dish are equal in strength.  Thus the effective collecting area $A_{\rm e}$ of the two-element interferometer equals the effective collecting area of each element. However, the noise voltages from the two interferometer elements are almost completely uncorrelated (only the point source contributes correlated noise), while the noise voltages going into the square-law detectors are completely correlated (identical).  In the limit where the antenna temperature $\Delta T$ contributed by the point source is much smaller than the system noise $T_{\rm sys}$, the correlator output noise is $2^{1/2}$ lower than the square-law detector noise from each antenna. For an unpolarized point source of flux-density $S$, $k \Delta T = S A_{\rm eff} / 2$ so for a single antenna
$$\sigma_{\rm S} = {2k T_{\rm sys} \over A_{\rm eff} (\Delta \nu_{\rm RF} \tau)^{1/2}}~$$ and for a two-element interferometer
$$\sigma_{\rm S} = {2^{1/2} k T_{\rm sys} \over A_{\rm eff} (\Delta \nu_{\rm RF} \tau)^{1/2}}~.$$ The point-source sensitivity of a two-element interferometer is therefore $2^{1/2}$ better than the sensitivity of each antenna, but $2^{1/2}$ worse than that of a single dish whose area is that of two antennas.  The reason the two-element interferometer is less sensitive than a single dish having the same total collecting area is that the information contained in the two independent square-law detector outputs has been discarded.  Together they have $2^{1/2}$ times the sensitivity of a single dish.  Combined with the independent correlator output, the total sensitivity is $(2 + 2)^{1/2} =$ twice the sensitivity of a single dish, or exactly the sensitivity of a single dish whose area equals the total area of the two-element interferometer.

The unsmoothed output voltage of a correlator whose inputs are uncorrelated Gaussian noise has a symmetric distribution with zero mean, and the rms fluctuation is a factor $2^{1/2}$ smaller than that of a square-law detector (see the corresponding figure in Section 3E).

The smoothed output voltage of a correlator approaches a Gaussian with zero mean, and the rms noise is reduced by the square root ot the number of independent samples averaged together.  This figure shows noise from an $N = 50$ sample running mean. The rms fluctuation is a factor $2^{1/2}$ smaller than that of a square-law detector (see the corresponding figure in Section 3E).

An interferometer with $N$ dishes contains $N(N-1)/2$ independent two-element interferometers, so its point-source sensitivity is
$$\bbox[border:3px blue solid,7pt]{\sigma_{\rm S} = {2 k T_{\rm sys} \over A_{\rm eff} [N(N-1)\Delta \nu_{\rm RF} \tau]^{1/2}}}\rlap{\quad \rm {(3G3)}}$$ In the limit of large $N$, $[N(N-1)]^{1/2} \rightarrow N$ and the point-source sensitivity of an interferometer approaches that of a single antenna whose area equals the total effective area $NA_{\rm eff}$ of the $N$ interferometer antennas.  For example, the VLA with $N = 27$ dishes each $d = 25$ m in diameter has the point-source sensitivity of a single dish whose diameter is $D = [N(N-1)]^{1/4}d = [27(26)]^{1/4}\times 25 {\rm ~m} = 129 {\rm ~m}$.  Had the square-law detector outputs been used as well, the point-source sensitivity of the $N$-element interferometer would be exactly the same as the sensitivity of a single dish having the same total collecting area.

Practical interferometers are slightly less sensitive than this because their correlators use digital multipliers that sample and quantize the input voltage, not perfect analog multipliers. For example, a multiplier that samples at twice the Nyquist rate with three quantization levels ($-1,0,+1$) is only 0.89 times as sensitive as a perfect multiplier.

Although the point-source sensitivity of an interferometer is comparable with the point-source sensitivity of a single dish having the same total area, beware that the brightness sensitivity of an interferometer is much worse because the synthesized beam solid angle of an interferometer is much smaller than the beam solid angle of a single dish of the same total effective area.  The angular resolution of an interferometer with maximum baseline $b$ is $\approx \lambda/b$ and the angular resolution of the single dish with diameter $D$ is $\approx \lambda/D$, so the beam solid angle of the interferometer is smaller by a factor $\approx (D/b)^2$.   This is roughly the filling factor of the interferometer, defined as the ratio of the area covered by all of the antennas to the area spanned by the interferometer array.  For example, the VLA in its $b \approx 11 {\rm ~km}$ "B" configuration has a filling factor $\approx (129 {\rm ~m} / 1.1 \times 10^4 {\rm ~m})^2 \approx 1.2 \times 10^{-4}$.  A high-resolution interferometer cannot detect a source of low surface brightness, no matter how high its total flux density.

The intensity axis of any astronomical image has dimensions of spectral brightness or specific intensity (e.g., units of Jy per beam solid angle or MJy sr$^{-1}$ or K), not flux density (e.g., Jy). The point-source rms $\sigma_{\rm S}$ in Equation 3G3 corresponds to image flux density per beam solid angle, e.g., Jy beam$^{-1}$.  Published radio images usually have intensity axes in units of Jy beam$^{-1}$ because the flux density of a point source equals its brightness in those units and because $\sigma_{\rm S}$ is independent of beam solid angle.  However, a proper spectral brightness depends only on the source. The "spectral brightness" specified in Jy beam$^{-1}$ has the dimensions of spectral brightness, but beware that this is not a proper spectral brightness because it depends on the synthesized beam solid angle and not just on the radio source.  Infrared astronomers frequently specify image intensity in MJy sr$^{-1}$, which is a proper brightness.  The brightness temperature $T$ is a convenient proper brightness for radio images.  The rms brightness-temperature sensitivity $\sigma_{\rm T}$ of an image made with beam solid angle $\Omega_{\rm A}$ follows directly from Equation 3G3 and the Rayleigh-Jeans law:
$$\bbox[border:3px blue solid,7pt]{\sigma_{\rm T} = \biggl({\sigma_{\rm S} \over \Omega_{\rm A}}\biggr) {\lambda^2 \over 2k}}\rlap{\quad \rm {(3G4)}}$$ Most interferometer images are made with Gaussian beams. The beam solid angle of a Gaussian beam with HPBW $\theta_0$ is
$$\Omega_{\rm A} = {\pi \theta_0^2 \over 4 \ln 2}~.$$ Beware that a high-resolution (low $\Omega_{\rm A}$) image with a good point-source sensitivity (low $\sigma_{\rm S}$) may still have a poor brightness-temperature sensitivity (high $\sigma_{\rm T}$).

Example: All of the 1.4 GHz NRAO VLA Sky Survey (NVSS) images were restored with a circular Gaussian beam having a $\theta_0 = 45$ arcsec half-power width.  What is the brightness temperature $T_{\rm b}$ corresponding to the rms noise $\sigma_{\rm S} = 0.45 {\rm ~mJy~beam}^{-1}$?

$$\Omega_{\rm A} = {\pi \theta_0^2 \over 4 \ln 2} \approx 5.39 \times 10^{-8} {\rm ~sr}~.$$ $$T_{\rm b} = \biggl({S \over \Omega_{\rm A}}\biggr) {c^2 \over 2 k \nu^2}$$ $$T_{\rm b} \approx \biggl({0.45 \times 10^{-29} {\rm ~W~m}^{-2}{\rm ~Hz}^{-1} \over 5.39 \times 10^{-8}{\rm ~sr}}\biggr) {(3\times 10^8 {\rm ~m~s}^{-1})^2 \over 2 \times 1.38 \times 10^{-23} {\rm ~J~K}^{-1} (1.4\times10^9 {\rm ~Hz})^2} \approx 0.14 {\rm ~K}$$

One 45 m dish of 30 comprising the Giant Metre-wave Radio Telescope (GMRT) near Pune, India.  The lightweight mesh surface and "rope trick" backup structure with tensioned cables is inexpensive and satisfactory for use at low frequencies ($\nu \leq 1.5$ GHz) in regions where ice loading is not a problem. Like the VLA, the GMRT is not confined to an east-west line and can make "snapshot" images. Image credit

The ten 25 m VLBA (Very Long Baseline Array) antennas combine to form an array 8000 km in size. Image credit

The Plateau de Bure interferometer of six 15 m antennas has an angular resolution of 0.5 arcsec at 230 GHz. Image credit

The Combined Array for Research in Millimeter-wave Astronomy (CARMA)
was created by moving the six 10-meter telescopes from Caltech's Owens Valley Radio Observatory and nine 6-meter telescopes from the Berkeley-Illinois-Maryland Association array to a new location at Cedar Flat in the Inyo Mountains near Bishop, CA

The Sub-Millimeter Array (SMA) consists of eight 6 m antennas capable of operating at wavelengths as short as 0.3 mm on Mauna Kea, HI. Image credit

Artist's conception of the Atacama Large Millimeter Array (ALMA) currently under construction at 5000 m elevation in northern Chile. Image credit

The Vertex test antenna for ALMA. Image credit

Artist's conception of the Square Kilometer Array ( SKA). Image credit