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--- un:radio-interferometer [2024/09/08 02:20] – created asad
+++ un:radio-interferometer [2025/11/21 22:34] (current) – asad
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 ====== Radio interferometer ======
+A two-element radio interferometer is the simplest building block of aperture-synthesis radio astronomy. It consists of two spatially separated antennas whose voltage outputs are multiplied and time-averaged by a *correlator*. Even very large arrays with \(N \gg 2\) antennas can be understood as a collection of \(N(N-1)/2\) independent two-element interferometers, each sampling one Fourier component of the sky brightness distribution.
+{{https://www.cv.nrao.edu/~sransom/web/x278.png?nolink&600}}
+Consider two identical antennas separated by a baseline vector \(\vec{b}\) of length \(b\), pointing toward a distant source in the direction of the unit vector \(\hat{s}\). If the angle between \(\vec{b}\) and \(\hat{s}\) is \(\theta\), then the plane wave from the source arrives at the two antennas at slightly different times. The extra path the wavefront travels to reach antenna 1 is
+$$
+\vec{b}\cdot\hat{s} = b\cos\theta,
+$$
+which introduces a **geometric delay**
+$$
+\tau_g = \frac{\vec{b}\cdot\hat{s}}{c}.
+$$
+If the interferometer is **quasi-monochromatic**, responding only to a very narrow bandwidth \(\Delta \nu \ll 2\pi/\tau_g\), then the signals received by the two antennas can be written as
+$$
+V_1(t) = V \cos[\omega(t - \tau_g)], \qquad
+V_2(t) = V \cos(\omega t),
+$$
+where \(\omega = 2\pi\nu\) is the angular frequency of observation and \(V\) is the voltage amplitude.
+The correlator multiplies the signals:
+$$
+V_1 V_2 = V^2 \cos[\omega(t - \tau_g)]\cos(\omega t)
+= \frac{V^2}{2}\left[\cos(2\omega t - \omega\tau_g) + \cos(\omega\tau_g)\right].
+$$
+A time average over many cycles removes the rapid oscillation at \(2\omega t\), leaving only the slowly varying term:
+$$
+R = \langle V_1 V_2 \rangle = \frac{V^2}{2}\cos(\omega\tau_g).
+$$
+This quantity \(R\) is the **correlator output**. Its amplitude is proportional to the source flux density \(S\) multiplied by the geometric mean of the antenna collecting areas \((A_1 A_2)^{1/2}\). Its phase encodes precise information about the source direction.
+Because the average of the product of uncorrelated noise is zero, the interferometer has **no DC output**. This suppresses:
+* receiver-gain fluctuations
+* very extended sky signals such as the CMB
+* short impulsive interference that does not reach both antennas at the same time
+This behaviour is fundamentally different from an adding interferometer (e.g., optical Michelson), which sums power rather than multiplying voltages.
+As the Earth rotates, the relative angle \(\theta\) changes, and the correlator output traces a **sinusoidal fringe**:
+$$
+R(t) = \frac{V^2}{2}\cos(\omega\tau_g(t)).
+$$
+The **fringe phase** is
+$$
+\phi = \omega\tau_g = \frac{\omega}{c} b \cos\theta,
+$$
+and its sensitivity to angle is
+$$
+\frac{d\phi}{d\theta} = -\frac{\omega b}{c}\sin\theta
+= -2\pi \left(\frac{b\sin\theta}{\lambda}\right).
+$$
+One full fringe cycle \(\Delta\phi = 2\pi\) corresponds to an angular shift
+$$
+\Delta\theta = \frac{\lambda}{b\sin\theta},
+$$
+which becomes extremely small when the projected baseline \(b\sin\theta\) spans many wavelengths. This makes interferometers exceptionally powerful for astrometry, routinely achieving absolute positional accuracies \(\sim10^{-3}\,\mathrm{arcsec}\) and differential accuracies \(\sim10^{-5}\,\mathrm{arcsec}\).
+If the antennas were isotropic, the interferometer would simply measure a pure sinusoid across the sky—one Fourier component of the brightness distribution. With directive antennas, this sinusoid is multiplied by the individual antenna voltage beams, producing a primary beam envelope (often Gaussian). The Fourier transform of this product corresponds to a finite range of angular spatial frequencies around \(b\sin\theta/\lambda\), set by the antenna diameter \(D\). Because \(D < b\), the interferometer cannot measure the lowest spatial frequencies and therefore cannot detect an isotropic sky component such as the 3-K cosmic microwave background. These missing short spacings can be filled by a sufficiently large single-dish telescope.
+Improving instantaneous imaging performance requires more baselines. An array of \(N\) antennas produces \(N(N-1)/2\) simultaneous baselines. The average of the point-source responses of all baselines produces the **dirty beam**, which approaches a Gaussian with full width \(\theta \simeq \lambda/b\) as the number of antennas grows. Missing short spacings produce negative sidelobe structures such as the characteristic “bowl.”
+Because sky brightness distributions usually do not change on observation timescales, one can use Earth-rotation synthesis or movable antennas to accumulate many baselines over time. A single two-element system with movable antennas could reproduce the baselines of a larger array by observing multiple spacings sequentially.
+{{:courses:ast301:two_element_interferometer.webp?nolink&650|}}
+===== Insights =====
+* A two-element interferometer measures the Fourier component of the sky brightness corresponding to its projected baseline.
+* The correlator output is proportional to \(\cos(\omega\tau_g)\); its phase encodes source position with extremely high precision.
+* No DC output means large-scale emission, receiver gain drifts, and uncorrelated noise are suppressed.
+* The fringe spacing \(\Delta\theta = \lambda/(b\sin\theta)\) sets the interferometer’s angular sensitivity.
+* Directive antennas impose a primary beam envelope on the fringes, limiting the range of spatial frequencies sampled.
+* Large arrays combine many baselines to synthesize a beam whose width is roughly \(\lambda/b\).
+* Missing short baselines cause negative bowls and sidelobes in the dirty beam; single-dish data can fill in these spacings.
+===== Inquiries =====
+- Explain how the geometric delay \(\tau_g\) arises from the baseline geometry of a two-element interferometer.
+- Derive the expression \(R = \frac{V^2}{2}\cos(\omega\tau_g)\) starting from the input voltages.
+- How does the Earth’s rotation generate interferometric fringes?
+- Why does a multiplying interferometer have no DC response, and why is this advantageous?
+- What spatial frequency of the sky brightness distribution is sampled by a baseline of length \(b\)?
+- Why can’t a two-element interferometer detect an isotropic signal such as the CMB?
+- How do missing short spacings influence interferometric imaging, and how can they be recovered?
+- Explain why interferometric astrometry can exceed the angular precision of individual telescope tracking.