1 Introduction

Orders of magnitude

• Radio wavelength
  • D=30 m, λ=1 mm: 1.22 λ/D ≈ 10"
  • Recall: 1 AU at 1 pc is 1" and 1 pc=2×10⁵ AU
  • Protostar: few 1000 AU at 150 pc, or θ=1000/150=6"
  • Protoplanetary disk: few 10 AU at 60 pc, or θ=0.2"
  • Molecular clouds in external galaxies: 10 pc at 10 Mpc or θ=2e6/1e7=0.2"
• Visible/IR
  • D=8 m, λ=0.5mic: 1.22 λ/D≈0.02"
  • Main sequence stars diameter 0.1 mas
  • inner disks: 1 AU at 60 pc, or θ=0.02"
• How to go beyond the diffraction limit ?
  • increase the diameter

• How to increase the sensitivity of a single-dish ?

  • increase the collecting area

ALMA interferometer

Protoplanetary disks observed with ALMA

IRAM/NOEMA interferometer

• Frequency range: 80 GHz to 350 GHz; baselines up to 1.5 km

ESO/VLT Interferometer

Applications of interferometry

• Measuring source sizes (e.g. Michelson)
• High angular resolution imaging (e.g. ALMA, or EHT)
• Astrometry (e.g. Very Long Baseline Interferometry)

Applications of interferometry

• Measuring source sizes (e.g. Michelson)
• High angular resolution imaging (e.g. ALMA, or EHT)
• Astrometry (e.g. Very Long Baseline Interferometry) and Celestial Frame Reference definition
• Geodesy

Optical vs Radio interferometry

• Different terminology suggests they are different in nature
• But fundamental principles are the same
• Yet, practical limitations make them different

Warning

• Notations will be largely inconsistent throughout because of using plots from various sources.
• Be careful with the notations for the diameter of each telescope and the baseline length

2 Basic principles

2.1 The Young interference and the twin-wave interferometer

• The Young's hole experiment: fringes produced by the interference between the coherent waves emitted by each hole
• The radio interferometer equivalent
  • two identical, coherent, transmitting antennas
  • used in emission, the electric field radiated coherently by each antenna interferes as in the Young's experiment
• The fringes can be 'seen' by looking at a point source, using the interferometer in reception: when the source is located at the position of a maxima, the output signal is maximum; as the source moves (e.g. because of Earth rotation), the signal would decrease, and increase again.

Twin-element interferometer

• 2 telescopes of diameters d separated by D
• Optical path difference (OPD): δ = Dsinθ
  • Geometric time delay: τ = Dsinθ/c
  • Phase difference: ψ = 2πν₀ τ = 2π Dsinθ/λ = 2πν₀ Dsinθ/c = kDsinθ, k=2π/λ
• Small angles θ: ψ ≈ kDθ

2.2 The additive interferometer

• In the classical Young's hole experiment, the electromagnetic field from each hole is summed and the resulting intensity is the square modulus of the combined field.
• Such a situation is called "additive interferometer"; this distinction is made because we shall see that modern radio interferometers are of a different kind, called "multiplicative interferometer".
• We first examine properties of the additive inteferometer.

The Michelson-Pease interferometer: measuring stellar diameters

• Measurement of a star diameter: αOrionis (Michelson & Pease 1921)
• Visibility (a contrast) is maximum, V=1, for a point-like source; V decreases when the source is resolved; it is zero when the source is fully resolved.
• V = (I_max-I_min)/(I_max+I_min)

Extended source seen with the twin-elements interferometer

• Consider two identical antenna separated by a distance B. Neglect the diameter of each antenna.
• Consider first a point source. The power from the additive interferometer is:
  • P(θ) = P₀ (1+cosψ) = P₀ [1+cos(kaθ)]
  • P₀: power detected by one antenna
• Now, consider an extended source of angular size 2W (≪ π) located at angle φ. The detected power is:
  • \(P(\theta) = \int_{\phi-W}^{\phi+W} P_0[1+\cos ka\theta]d\theta = P_0[1+\cos(\psi) \sin(kaW)/kaW]\),
  • P₀=2pW is the power from the whole source on each antenna;
  • As the source moves, the output power still oscillates between a maximum and a minimum, but the amplitude of the oscillation is reduced by the source extension

• P(θ)= P₀[1+cos(kaθ) sin(kaW)/(kaW)]
  • P_max and P_min depend on the source extent W: R=P_min/P_max: fringes disappear when R=1;
• Visibility: V=|P_max-P_min|/(P_max+P_min)| = |sinc(kaW)|
  • V=0 for W=n λ/2D, n∈ Z: P_max=P_min=P₀
  • V=1 (maximum) for a point source: P_max = 2P₀. P_min=0;
• Conversely, by modifying the separation \(a\) between the two telescopes, measuring the visibility may provide W, the source size (Michelson experiment)

2.3 Early additive radio inteferometers

The first radio interferometer

February 7th 1946, Dover Heights, Sydney, McCready, Pawsey, and Payne-Scott 1947

Early additive interferometer

• Drift interferometer: Ryle and Vonberg 1946; dipole antenna arrays at 175 MHz; separation from 10 to 140λ (17 to 240 m)
• Adding and squaring: P = (V₁+V₂)² = V²[1+cos(ψ)] + high-freq terms, ψ=2πν₀ τ
• dτ/dt due to Earth rotation is small: ψ varies slowly; apply low-pass filtering (time averaging over Δ t ≫ 1/ν₀)
  • P = P₀ (1+cosψ), P₀=power from one telescope
  • P varies from 0 to 2P₀

• Interpretation: cos(ψ) = Fourier component of the source brightness of angular frequency ψ
• The inteferometer is sensitive to this Fourier component
• As we will now see, the shape of the response is the cosine modulated by primary beam pattern, the beam pattern of each telescope.

2.4 Fourier space view of the additive interferometer

Recall the general relationships between the various functions:

Fourier space view

• Notation: separation s_λ=D/λ, diameter a_λ=d/λ
• Primary pattern: E_one(θ) = sin(πθ a_λ)/(πθ a_λ); first null for θ=1/a_λ
• Array pattern: cos(ψ/2) = cos(πθ D_l); TF[cos(2π u₀ θ)](u)=1/2 [δ(u-u₀) + δ(u+u₀)]; period=2/s_λ

• Interferometer Power pattern: P(θ) = |E_one(θ)|² cos²(ψ/2) = |E_one(θ)|² (1+cos2ψ) = sinc²(πθ a_λ) cos(2πθ D_λ)
  • first nulls at ±1/2a_λ: angular resolution ∼ λ/a
• Direct space: auto-correlation or transfer function is Low-pass + Pass-band filter

Visibility of the additive interferometer

• Convolve the interferometer power pattern by the source brightness distribution
  • Point source: recover the interferometer power pattern
  • As the source extent increases, blurring of the fringes
  • Visibility decreases as the source becomes spatially resolved
• Visibility is related to the flux density: V=|P_max-P_min|/(P_max+P_min)
• Visibility is measured from the observed intensity record

Problems with the additive interferometer

• P = P₀ + P_0cosψ + P_noise
  • The output of the interferometer contains dominant noise power contributions: galactic background, ground, amplifiers
  • The amplitude of this noise power is much larger (several orders of magnitude) than P₀
  • Furthermore, P_noise is, like P₀, subject to changes in receiver gain: drifts and visibility smearing

3 Multiplicative interferometers

• Great advance: phase-switching interferometer
• Periodic (∼ few 10 Hz) phase reversal of signal 2: measure alternatively \((V_1+V_2)^2\) and \((V_1-V_2)^2\)
• Time-averaged difference: (V₁+V₂)² - (V₁-V₂)² ∝ <V₁ V₂ >
• Output F ∝ V² cos2πν₀ τ = V² cos(ψ)
• cosψ: Fourier component of the source brightness, as for the additive interferometer

Phase-switched interferometer

• Phase-switching interferometer: Ryle 1952
• Ryle & Hewish: 1974 nobel prize
• Yet, now, source confusion becomes a problem! Fringes from nearby sources may overlap.

Fourier analysis

• One antenna has a π/2 phase difference half of the time
• Individual normalized field pattern: \(f_{\rm one}(\theta)\)
• Power pattern: \(P(\theta) = |f_{\rm one}(\theta)|^2 \cos2\pi\nu D\sin\theta/c\)
• Multiplicative interferometer = pass-band filter

Advantages of the multiplying interferometer

• Low-pass filter removed: Time-varying constant \(V^2\) is removed
• Reduce sensitivity to instrumental gain variations: amplifier at each antenna so longer baselines

Multi-antenna interferometer

Synthesis imaging

• One baseline is sensitive to one Fourier component
• Multiple telescopes allow to measure several Fourier components simultaneoulsy
• Actually, the angular frequency of source brightness that is measured is set by the baseline projected perpendicular to the line of sight
• Modifying the projection angle amounts to measure different components: use Earth rotation to synthesize a virtual aperture
• Picture: source at high declination; general case, ellipses

3.1 Frequency filtering

Low-pass vs Pass-band filters

• Left to right
  • Filled aperture: low-pass
  • Additive: low-pass + band-pass
  • Multiplicative: band-pass

Zero-spacings

• Multiplicative interferometer = band-pass, centered on b cosθ/λ angular frequency
• antenna can not be packed closer than their diameter D: b>D so that angular frequencies lower than D/λ can not be measured with such an interferometer
• However, their could be some important flux at low angular frequency: provided by single-dish telescopes with D_single > D_interfero.

3.2 Modern radio interferometers

• Output voltages are amplified (H1, H2), multiplied, and integrated (sumed) over a time interval 2T (∼ msec to sec):
• τ=geometrical + instrumental time delay = 2πν Dsinθ/c + τ_i
• Correlator output has dimensions of (voltage)²:
  • V₁ ⋅ V₂ = v² cos(ω t) cos[ω(t-τ)] = v²/2 [cos(2ω t - ωτ) + cos(ωτ)]
  • r = <V₁(t)V₂(t-τ_g)>_2T = v₁ v₂ cos(ψ) = v₁ v₂ cos[2πν₀ τ_g(t)]
  • v₁, v₂, τ: vary on timescale >> 1/ν₀
  • Earth rotation: τ(t), hence r(t) oscillates: fringe pattern

• Multiplicative is performed directly in a correlator; synchroneous phase-switching no longer needed
• Amplification at each antenna allows long baselines (13km for ALMA)
• Fringe phase: ψ = ωτ; fringe period corresponds to angular shift λ/b_proj, b_proj is the projected baseline as seen from the source at zenith angle θ

Cygnus A: from early to modern interferometers

• Cygnus A: bright Active Galactic Nucleus (AGN); energy released by gas in-fall onto SMBH
• At scales <1pc, accretion disk around SMBH: X-ray and UV emission
• Scales 1-100 pc: obscuring torus. In-fall in SMBH potential well; toroidal distribution, seen through infrared emission by warm dust

4 What do radio interferometers measure ?

• Interferometers are 'mapping machines': the reason is that they measure the Fourier transform of the sky brightness;
• Interferometers measure some Fourier components of the brightness, but not all (recall that they are bandpass filters — plus a low-pass for additive interferometers);
  • one Fourier component is one cosψ term, with ψ=phase difference, for a given direction, between a pair of telescopes;
  • thus, if N telescopes, N(N-1)/2 Fourier components are measured at any given time;
  • output is V² cosψ (multiplicative) or V²(1+cos 2ψ) (additive)
• The Fourier transform is measured within a given area on the sky centered around the reference phase center;

• Two key concepts:
  • Primary beam: this is the beam of a single telescope (of diameter D), ≈ λ/D
  • Synthesized beam: this is the beam after all telescopes have been combined, ≈ λ/B, with B∼ maximum baseline
• The interferometer records the Fourier components (called visibilities) of I within the primary beam; hence, the primary beam defines the maximum FoV of the interferometer.
• To map areas larger than the primary beam: mosaicing (several reference phase centers);

4.1 Extended source and the complex visibility

• Observing an extended source, with phase reference at \(\vec{s}_0\): \(\vec{s}=\vec{s}_0+\vec{\sigma}\)
• Power at each antenna: 1/2 A(σ) I_ν(σ) Δν dΩ
  • A(σ) = single-antenna power pattern toward direction σ
  • I_ν(σ) = brightness from direction σ
• Interferometer response: this multiplied by fringe pattern cos(ωτ_g):
• Geometrical delay: τ_g=\(\vec{b}\cdot\vec{s}/c\)

• Correlator output from dΩ, within dν:

  \(dr = \frac{1}{2} A(\vec{\sigma})I(\vec{\sigma})\cos(\omega\tau_g)d\Omega d\nu\)

• We have obtained the
• We will now introduce the complex visibility

Extended source

• Assume that effective area A(ω)=A_e P(ω) is identical for each antenna and that source is incoherent (add the power from different regions over the source);

• Correlator output from all (incoherent) source elements:

  \[r = 1/2 d\nu\int_{4\pi} A(\vec{\sigma})I(\vec{\sigma})\cos(2\pi \vec{b}\cdot\vec{s}/\lambda)d\Omega\]

• Now: \(\vec{s}=\vec{s}_0+\vec{\sigma}\), with \(\vec{s_0}\) the phase center position (or tracking position) and \(\vec{\sigma}\), the position offset measured wrt \(\vec{s_0}\);
  • note τ₀(t) = b⋅ s₀(t) /λ;
  • \(\cos(2\pi b\cdot s/\lambda) = \cos(2\pi\nu_0 \tau_0 + 2\pi\nu_0 b\cdot \sigma) = \cos()\cos()-\sin()\sin()\)

• We define the complex visibility by:

  \[ {\cal V} = |{\cal V}|e^{i\Phi_{\cal V}} = \frac{1}{P(0)} \int_{4\pi} A(\vec{\sigma})I(\vec{\sigma}) e^{-i 2\pi\nu_0 \vec{b}\cdot\vec{\sigma}}d\Omega \]

• Correlator output can then be written:

  2r(t)/dν = cos(2πτ_g ν₀) Re(V) + sin(2πτ_g ν₀) Im(V) = |V| cos[2πν₀ τ₀(t) - Φ_V]

4.2 Effects of finite bandwidth and delay tracking

Bandwidth pattern

• Receivers have a finite bandwidth (ν=ν₀ ± Δν/2): the fringe pattern cos(ωτ) is multiplied by the 'envelope' (FT of H-function is a sinc)

  F_B(θ)=sin(πΔντ_g)/(πΔντ_g), τ_g=b sinθ/c≈ bθ/c

• Output of correlator integrated over Δν is thus: \(R(t) = \frac{1}{\Delta\nu} \int r(t) = \frac{1}{2} |V|\cos[2\pi\nu_0 \tau_0(t) - \Phi_V] F_B(\theta)\)
• F_B(θ) has a first zero at Δντ=1 or θ≈ c/2π DΔν; 10% loss in fringe intensity is reached with a zenith angle of 5' for a bandwidth of 500 MHz and a baseline b=100m.
• Observing a source for long integration times requires geometrical delay compensation

Instrumental delay τ_I and phase reference center

4.3 Visibility in the (u,v) plane

• (u,v,w): coordinates of the baseline b (in units of λ), in a right handed coord. system where w is along s₀ and (u,v) perp. \(\vec{s_0}\);
• (x,y,z): coord. of the source vector \(\vec{s}\) in this frame (also called direction cosine and usually noted (l,m,n))
  • \(\vec{s}\) is unitary: x² + y² + z² = 1
• \(\vec{b}\cdot\vec{s}_0/\lambda = w\)
• \(\vec{b}\cdot\vec{s}/\lambda = ux + vy + wz\), \(z^2=1-x^2-y^2\)
• dΩ=dx dy/z=dx dy/(1-x²-y²)^1/2

• Visibility:

  \(V(u,v,w) = \frac{1}{\sqrt{1-x^2-y^2}} \int\int A(x,y)I(x,y)e^{-i2\pi(ux+vy)}e^{-i\pi(x^2+y^2)w}dxdy\)

What are visibilities?

• If the field of view is small enough, \(\pi(x^2+y^2)w\) can be neglected, this reduces to a 2D Fourier transform:

  \(V(u,v) \approx V(u,v,0) = \int_{-\infty}^\infty\int_{-\infty}^\infty \frac{A(x,y)I(x,y)}{\sqrt{1-x^2-y^2}}e^{-i2\pi(ux+vy)}dxdy\)

• Define A'(x,y) = A(x,y)/(1-x²-y²)^1/2
• Then, visibilities are the FT of A'× I, the primary beam multiplied by the source brightness distribution
• Conversely: \[ \frac{A(x,y)I(x,y)}{\sqrt{1-x^2-y^2}} = \int_{-\infty}^\infty\int_{-\infty}^\infty V(u,v)e^{i2\pi(ux+vy)}dxdy \]

• \(|{\cal V}|\): fringe amplitude or visibility
• \(\Phi_{\cal V}\): fringe phase
• \([{\cal V}]\) = W m^-2 Hz^-1

4.4 Summary

• Radio interferometers measure so-called complex visibilities
• V(u,v,w) is related to the FT of the source;
• V(u,v) = Fourier component of the source brightness weighted by antenna primary beam
• Each baseline provides one Fourier component in the \((u,v)\) plane
  • Because I(x,y) is real, V is hermitian: V(u,v) = V^*(-u,-v)
  • So each measurement provides two visibilities
  • N antenna deliver N(N-1)/2 independent instantaneous baselines
  • N antenna thus give N(N-1) visibilities
• Spatial frequency: baseline projected on the source, perp. s₀

5 Imaging with interferometer

• Interferometer intrinsically makes images as soon as the source is resolved
• However, the images are in the Fourier domain ((u,v) plane): the outputs are the visibilities
• Reconstructing the direct image from its Fourier transform is done through deconvolution
• Problem: the (u,v) plane is only partially sampled; not all Fourier components are measured
• Several methods to perform deconvolution (often the time-consuming part)

5.1 (u,v) coverage

• The ideal interferometer achieves the angular resolution of a filled aperture with diameter equal to the largest baseline D_max
• One pair of antenna provides only two visibilities at a given time: this is just as if you were covering a filled aperture of diameter D_max with an opaque screen, leaving only two holes separated by the baseline.
• The beam pattern (or PSF) of such a 'two-hole' entrance pupil is what is called the dirty beam. Just as the beam pattern is the Fourier Transform of the aperture function, the dirty beam will be the FT of the two holes, that is a 'sampling' function made of two δ functions, convolved, by a circular aperture.
• Let us explore what does the dirty beam looks like, and how people have managed to obtain dirty beams that are good enough for robust imaging.

Image synthesis

• Aim: measure as much V(u,v) as possible, i.e. obtain dense sampling of the (u,v) plane;
  • 1 pair of telescope: 1 (u,v) sample at a given time
  • N telescopes: N(N-1) visibilities at a given time
  • Two solutions:
    • Use different telescope configurations
    • Use super synthesis: use Earth rotation to measure different Fourier components; projected baseline changes with time, corresponding to different points in the (u,v) plane

Dirty beam: 2 antenna

taken from Lecture by D. Wilner

Note the reversed x-axis in the direct (dirty beam) plane

• δ(u,v) -> 1
• shift theorem: if F(x)=TF[f(u)] then TF[f(u-u₀)]=e^{-2iπ u₀ x} F(x)
  • In 2D: TF[f(u-u_0,v-v₀)] = e^{-2iπ(u₀ x + v₀ y)} F(x,y)
  • Applied to δ(u,v): ∝ cos[2π(xu₀ + yv₀)]
  • Lines of constant amplitude: xu₀ + yv₀ = cst
  • Maxima are lines inclined by an angle atan(-u₀/v₀) = π/2+atan(v₀/u₀)
• Quickly, as N increases, it becomes impossible to anticipate the shape of the dirty beam.

Image Synthesis: increasing the number of antenna

Adapted from Lecture by D. Wilner

Super Synthesis: increasing the number of samples (Earth rotation)

Adapted from Lecture by D. Wilner

Field of view and zero-spacings

• Primary beam multiplication
• Zero-spacings

Summary

• The better the (u,v) coverage, the better the image reconstruction
• Visibilities = FT(A.I)
• In practice:
  • Incomplete sampling: V = Σ(u,v) × FT(A ⋅ I)
  • Noise: V = Σ(u,v) × FT(A ⋅ I) + N
• Inverse Fourier transform of the complex visibility:
  • F = FT^-1(V) = D * (A ⋅ I) + N
    • F: Dirty map
    • D: dirty beam = FT^-1 (Σ)
    • N: noise
  • Incomplete uv coverage: strong sidelobes in D
• Image making needs: from dirty map to clean map (algorithms)

5.2 Deconvolution

• the more complete the uv-coverage, the easiest and more robust the deconvolution
• deconvolution efficiency depends on the noise level