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• Outline
• Glossary
• 4. The Visibility space
  • Previous: 4.5.2 UV Coverage
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4.6 The Fourier Approximation & the van Cittert-Zernike Theorem

An interferometer does not measure the sky, but rather its associated visibility function. Even in the absence of distortions between the sky signal and our measurement thereof, we still need to simplify our framework to perform aperture synthesis.

The Van Cittert-Zernike theorem makes the link between sky and visibility function explicit. So far, we have simply stated that there was a Fourier relation between what our antennas measure and the sky itself. Our aim is now to show the exact regime in which this is true.

In $\S$ 4.6.1 ⤵, we will start by taking some approximation which will simplify Eq. 4.4.2 ➞ in $\S$ 4.4 ➞.

4.6.1 Fourier approximations

Before stating the theorem, we need to make several approximations to allow its formulation. Sections $\S$ 4.6.1.1 to $\S$ 4.6.1.7 cover these various approximations, from the need for a distant source to the requirements on the observer's local conditionts.

4.6.1.1 Mutual incoherence of the source

This approximation consists of assuming that the different parts of the sky are statistically independent from each other. The mutual coherency between two direction is then 0. The sky can be described as a collection of point sources. Therefore, the visibility function is linear with respect to the directions in the sky, as the correlation between $\mathbf{s}_i$ and $\mathbf{s}_j$ is non zero only when $i=j$.

4.6.1.2 Far-field approximation

In the ideal case, we are observing infinitely-distant sources that lie on the celestial sphere. The antenna patterns are then adequately described by the Far-Field pattern. The visibility expressed in spherical coordinates is therefore independent of the distance from the observer to the source, $r$. We assume that the distance $R >> \frac{|\mathbf{b}_\text{max}|^2}{\lambda}$ such that the dependence of Eq. ?? TLG:RC: Fix Eq link. is simplified. All extended sources are deformed due to the celestial sphere's projection onto our 2D plane.

4.6.1.3 Homogeneity of the propagation medium

In the simple case of classical radio sources, we assume that no propagation effects affect the signal (e.g. absorption, refraction, distortion). This is not the case when observing:

• a signal polarised by passing through a magnetized medium (e.g. the Faraday rotation of polarized radio emission through the magnetized IGM/ISM)

• a pulsar seen through the Intra Galactic Medium (IGM), which causes signal distortions (spectral dispersion and temporal broadening of pulses)

4.6.1.4 Small-field approximation

Real-life antennas are not isotropic. They are sensitive to a limited fraction of the sky, which is defined by their Field of View. Close to the phase center, the celestial sphere can be approximated by its tangent plane. The sources in our FoV are then treated as though being on a plane.

Assuming that the observing region of the sky is small, we will consider that:

• the antenna gain is considered constant over the FoV: $A(l,m) \approx const.$
• the sky is projected onto the $(l,m)$ plane: $(\alpha_0,\delta_0)\leftrightarrow(l,m)$
• the extended sources are small relative to the FoV: $\Delta\theta_\text{source} << \Delta\theta_\text{FoV}$
• $w$ is small and $n$ is close to unity (see $\S$ 4.6.1.E) TLG:RF: Fix equation reference.

4.6.1.5 Coplanar baselines

In Eq. 4.4.2 ➞ in $\S$ 4.4 ➞, the exponent factor of the intensity distribution in the expression of the visibility function is $K(l,m)$, defined as:

$$K(l,m)=\exp \left[ {-\imath 2\pi (ul+vm+w(\sqrt{1-l^2-m^2}-1)) }\right]$$

For a tracking interferometer (which compensates for the phase delay between the reception of the same signal by different antennas), and with the small-field approximation, we measure spatially incoherent emission as though it was sent from sources on the ($l$,$m$) plane, close to the phase center at ($l=0$,$m=0$).

If ($l$,$m$)$<<$1 then $w(\sqrt{1-l^2-m^2}-1) \rightarrow 0$.

In that case, the $w$-term can be ignored and $K(l,m)$ reduces to $$K(l,m)=\exp \left[ {-\imath 2\pi (ul+vm) }\right]$$

4.6.1.6 Narrowband approximation

The correlator operates in a narrow bandwith $\Delta \nu$ (condition). The bandwidth will modulate the width of the fringe damping factor by a sinc function. If we limit ourselves to the narrow-band case, we can assume that the the sinc damping our fringe is effectively infinitely wide.

However, we usually work with some bandwidth $\Delta \nu$. The effect of the bandwidth pattern on the interferometer response can thus be limited. We do this by constraining the observed field to a region around the phase center where spatial coherency is maintained: $$ \frac{\Delta\nu}{\nu} < \frac{1}{l_\text{max} u}, \frac{1}{m_\text{max} v} $$

The equivalent effect in broadlight interferometry is the disappearance of fringes as distance from fringe centre increases.

4.6.1.7 Continuous sampling approximation

We assume that our sampling of the complex visibility function is continuous, and that we therefore do not suffer from the sampling effect in the image pace (i.e. we do not experience any effect due to the sampling itself). This effect is a major challenge to deal with, and will be described in more detail in the imaging chapter ($\S$ 5. ➞).

4.6.2 Van Cittert-Zernike Theorem

Originally formulated in the optical regime, this theorem links the intensity distribution of the sky with the cross-correlation of the signals received by two receivers $R_1$ and $R_2$. Given the assumptions in $\S$ 4.6.1 ⤵, we are able to state the Van Cittert-Zernike Theorem.

4.6.2.1 Formulation

Given an extended, monochromatic and incoherent intensity distribution $I_\nu$, the theorem states that the complex visibility function associated with (i.e. measured by) a baseline $pq$ is linked to the intensity distribution via a 2D Fourier Transform:

$$\boxed{\boxed{V_{pq}(u,v,0)=\int_{-\infty}^\infty\int_{-\infty}^\infty{I_\nu e^{-2\imath\pi (ul+vm)}dldm}}}$$

4.6.2.2 Demonstration

Let us assume that our sky consists of a monochromatic, extended, incoherent intensity distribution.

The mutual coherence between the measurements of the signal from a distant source at two locations $R_1$ and $R_2$, over some amount of time $T$, is: $$\Gamma_\text{12}(u,v,\tau)= \lim_{T \rightarrow \infty} \frac{1}{2T} \int_{-T}^{T} E_1(t)E_{2}^{*}(t-\tau) dt$$

where the retarded electric fields can be expressed as TLG:AC: Give a reference where this can be verified.:

$$E_1(l,m,t)= E_{0}(l,m,t-\frac{r_1}{c}) \frac{e^{-2\imath\pi \nu(t-\frac{r_1}{c})}}{r_1}$$$$E_2(l,m,t)= E_{0}(l,m,t-\frac{r_2}{c}) \frac{e^{-2\imath\pi \nu(t-\frac{r_2}{c})}}{r_2}$$

where $r_1$ and $r_2$ are the distance of each receiver from the source.

Correlation of $\mathbf{E_1}$ and $\mathbf{E_2}$

\begin{eqnarray} \langle E_1(l_i,m_i,t) E_2^*(l_j,m_j,t) \rangle &=& \langle E_{0} (l_i,m_i,t-\frac{r_1}{c}) E_{0}(l_j,m_j,t-\frac{r_2}{c}) \rangle \frac{e^{-2\imath\pi \nu(t-\frac{r_1}{c})} e^{+2\imath\pi \nu(t-\frac{r_2}{c})}}{r_1 r_2} \\ &=& \langle E_{0}(l_i,m_i,t') E_{0}(l_j,m_j,t'-\frac{r_2-r_1}{c}) \rangle \frac{e^{2\imath\pi \nu \frac{r_1 - r_2}{c}}}{r_1 r_2}\\ &\approx& \langle E_{0}(l,m,t')E_{0}(l,m,t') \rangle \frac{e^{2\imath\pi \nu \frac{r_1 - r_2}{c}}}{r_1 r_2}\\ \end{eqnarray}

• Here, the incoherent sky approximation means $\langle E_{0}(l_i,m_i,t) E_{0}(l_j,m_j,t-\frac{r_2-r_1}{c}) \rangle = 0$ for $i \neq j$.

• In the far-field and small-FoV (Field of View) regimes, we have $r_1 \approx r_2 \approx r$. $(r_1 - r_2)/c$ can thus be approximated to zero: the time retardation is small enough to be negligible.

We can now determine the expression for the mutual coherence $\Gamma_{12}$:

$r_1-r_2$ represents the Optical Path Difference (OPD) between the two receivers, and can be expressed as a function of ($u$, $v$) and ($l$, $m$): $$r_1-r_2= \frac{c}{\nu} (u_{12}l+v_{12}m)$$

$$\Gamma_{12}(u,v,\tau=0)= \int_{\text{Source}} { \frac{I_\nu e^{2\imath\pi \nu \frac{(r_1-r_2)}{c}}}{r_1 r_2} }ds = \int_{\text{Source}}{\frac{I_\nu e^{2\imath\pi (u_{12}l+v_{12}m)}}{r^2}} ds$$

As ($l$,$m$) are direction cosines, they can be linked to spherical coordinates. On the projected sphere, the surface element $ds$ can be expressed as: $$ds = r^2 \; dl \; dm$$

The integrand is bounded to zero: temporal coherency limits and antenna response are such that the measured signal from increasingly distant sources will tend towards zero. We can thus extend the bounds of the integral to infinity with no loss of generality.

$\Gamma_{12}$ becomes the complex visibility $V_\nu$

$$ \Gamma_{12}(u,v,\tau=0) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} { I_\nu(l,m) e^{-2\imath\pi(u_{12}l+v_{12}m)}} dl dm =V_{12}(u,v,0) $$

In this form, the complex visibility $V_{12}$ is the Fourier transform of the intensity distribution $I_{\nu}(l,m)$. TLG:RC: Why this switch? EB: some maths I assume. Will look into it.

This theorem (and its link with Fraunhofer diffraction in the classical regime) are explained in greater detail in Thompson, Moran and Swenson $\S$ 14, p. 594 ⤴.

This theorem can also be derived directly from the RIME framework $\S$ 7.2 ➞. Bear in mind that we do not sample all the points in the $uv$ plane, and that a lot of information is missing as a consequence. Within the scope of our simplifying assumptions, it is nevertheless possible to recover the intensity distribution $I_{\nu}$ from partial measurements. Indeed, it actually consists of solving an inverse problem. In other words, the need for deconvolution techniques arises because we only have partial information on the full visibility function. TLG:RC: Rewrite prev paragraph. EB: it looks fine to me?

4.6.3 Concluding remarks

Radio interferometry consists of combining cross-correlation of signals measured with individual antennas, converting incoming incoherent EM waves into voltages. In order to make these signals "interfere", we devise a system to combine them in phase. To do so, we introduce the concepts of baseline, projected baseline and phase center. TLG:RC: Rewrite prev paragraph. EB: do not see how to phrase it more clearly

We started in $\S$ 4.2 ➞, by applying the tools introduced in $\S$ 3 ➞ to express the baseline in a convenient astronomical frame of reference, attached to the celestial sphere. In $\S$ 4.3 ➞, we studied the simple case of a 2-element interferometer, for which we derived the relevant interferometric quantities (phase center, coherence, visibility).

TLG:GM: Are terms in glossary.

In $\S$ 4.4 ➞, we extended the 1D case to a 3D definition of the visibility function. To give a physical interpretation of the relevance of the visibility function to study celestial objects, we linked the projected baseline to a spatial filter applied to the sky. With this, we demonstrated that the shape and position of a source could be accurately measured by an interferometer ($\S$ 4.4.2 ➞).

An interferometer samples the visibility function of the sky in a non-random manner. From the geometry of the interferometer with respect to the observed source, it was possible in $\S$ 4.5.1 ➞ to derive the actual path taken in the $uv$-space by a physical baseline over the course of an observation: they follow elliptical paths, which can potentially be used to increase the instrument's coverage of the $uv$-plane. The more diversified and numerous the samples in the $uv$ plane, the better our knowledge of the visibility function. In order to improves the $uv$ coverage, $\S$ 4.5.2 ➞ showed three ways to increase an interferometric instrument's $uv$-coverage.

The collection of $uv$ samples can be used to retrieve an approximation of the sky. Under a series of approximations $\S$ 4.6.1 ⤵, the definition of the visibility function can be linked to the intensity distribution through a Fourier transformation $\S$ 4.6.2 ⤵.

Having established this relationship, we can use the Van Cittert-Zernike theorem to cast the imaging problem as an inverse problem: to recover the sky $\mathcal{I}_\nu$ from an incomplete knowledge of its Fourier Transform, i.e. the sampled visibility function $\mathcal{V}_\nu$.

In $\S$ 5 ➞, we will discuss the principles of imaging with the Fourier Transform. Then, in $\S$ 6 ➞, we cover the various techniques used to perform image deconvolution, i.e. solve the inverse problem of imaging by aperture synthesis. TLG:GM: Are terms in glossary.

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