The Van Cittert Zernike Theorem

The van Cittert-Zernike theorem relates the spatial coherence function $V({\bf r}_1,{\bf r}_2) = \bigl<{\rm E}({\rm r}_1)E^*({\rm r}_2) \bigr>$ to the distribution of intensity of the incoming radiation, $\mathcal{I}({\bf s})$. It shows that the spatial correlation function $V({\bf r}_1,{\bf r}_2)$ depends only on ${\bf r}_1-{\bf r}_2$ and that if all the measurements are in a plane, then

$$V({\bf r}_1,{\bf r}_2) = \mathcal{F} \{I({\bf s})\}$$ (2.4.4)

where $\mathcal{F}$ implies taking the Fourier transform. Proof of the van Cittert-Zernike theorem can be found in a number of textbooks, eg. ``Optics'' by Born and Wolf, ``Statistical Optics'' by Goodman, ``Interferometry and Synthesis in radio astronomy'' by Thompson et al. We give here only a rough proof to illustrate the basic ideas.

Let us assume that the source is distant and can be approximated as a brightness distribution on the celestial sphere of radius $R$ (see Figure 2.2). Let the electric field^2.5 at a point $P_1^{\prime}(x_1^{\prime},y_1^{\prime},z_1^{\prime})$ at the source be given by $\mathcal{E}(P_1^{\prime})$. The field $E(P_1)$ at the observation point $P_1(x_1,y_1,z_1)$ is given by^2.6

Figure 2.2: Geometry for the van Cittert-Zernike theorem

$$E(P_1) = \int \mathcal{E}(P_1^{\prime}) { e^{-ikD(P_1^{\prime},P_1)}\over D(P_1^{\prime},P_1)}d\Omega_1$$ (2.4.5)

where $D(P_1^{\prime},P_1)$ is the distance between $P_1^{\prime}$ and $P_1$. Similarly if $E(P_2)$ is the field at some other observing point $P_2(x_2,y_2,z_2)$ then the cross-correlation between these two fields is given by

$$\bigl<E(P_1)E^*(P_2)\bigr> = \int \bigl<\mathcal{E}(P_1^{\pr... ...ver D(P_1^{\prime},P_1)D(P_2^{\prime},P_2)}d\Omega_1 d\Omega_2$$ (2.4.6)

If we further assume that the emission from the source is spatially incoherent, i.e. that $\bigl<\mathcal{E}(P_1^{\prime})\mathcal{E}^*(P_2^{\prime})\bigr> = 0$ except when $P_1^{\prime}= P_2^{\prime}$, then we have

$$\bigl<E(P_1)E^*(P_2)\bigr> = \int \mathcal{I}(P_1^{\prime}) ... ...},P_2)]}\over D(P_1^{\prime},P_1)D(P_1^{\prime},P_2)}d\Omega_1$$ (2.4.7)

where $\mathcal{I}(P_1^{\prime})$ is the intensity at the point $P_1^{\prime}$. Since we have assumed that the source can be approximated as lying on a celestial sphere of radius $R$ we have $x_1^{\prime}= R\cos(\theta_x) = Rl$, $y_1^{\prime}= R\cos(\theta_y) = Rm$, and $z_1^{\prime}= R\cos(\theta_z) = Rn$; ($l, m, n$) are called ``direction cosines''. It can be easily shown^2.7 that $l^2+m^2+n^2=1$ and that $d\Omega = {dl\ dm \over \sqrt{1-l^2-m^2}}$. We then have:

$$D(P_1^{\prime},P_1) = \bigl[(x_1^{\prime}- x_1)^2 + (y_1^{\prime}- y_1)^2 + (z_1^{\prime}- z_1)^2 \bigl]^{1/2}$$ (2.4.8)

$$= \bigl[(Rl - x_1)^2 + (Rm - y_1)^2 + (Rn - z_1)^2 \bigl]^{1/2}$$ (2.4.9)

$$= R\bigl[(l - x_1/R)^2 + (m - y_1/R)^2 + (n - z_1/R)^2 \bigl]^{1/2}$$ (2.4.10)

$$\simeq R\bigl[(l^2 +m^2+n^2) -2/R(lx_1+ my_1 +nz_1) \bigl]^{1/2}$$ (2.4.11)

$$\simeq R - (lx_1+ my_1 +nz_1)$$ (2.4.12)

Putting this back into equation 2.4.7 we get

$$\bigl<E(P_1)E^*(P_2)\bigr> = {1 \over R^2} \int \mathcal{I}(... ...-x_1)+m(y_2-y_1)+n(z_2-z_1)]} {dl\ dm \over \sqrt{1-l^2-m^2}}$$ (2.4.13)

Note that since $l^2+m^2+n^2=1$, the two directions cosines $(l,m)$ are sufficient to uniquely specify any given point on the celestial sphere, which is why the intensity $\mathcal{I}$ has been written out as a function of $(l,m)$ only. It is customary to measure distances in the observing plane in units of the wavelength $\lambda$, and to define ``baseline co-ordinates'' $u,v,w$ such that $u=(x_2 - x_1)/\lambda$, $v=(y_2 - y_1)/\lambda$, and $w=(z_2 - z_1)/\lambda$. The spatial correlation function $\bigl<E(P_1)E^*(P_2)\bigr>$ is also referred to as the ``visibility'' $\mathcal{V}(u,v,w)$. Apart from the constant factor $1/R^2$ (which we will ignore hence forth) equation 2.4.14 can then be written as

$$\mathcal{V}(u,v,w) = \int \mathcal{I}(l,m) e^{-i2\pi[lu+mv+nw]} {dl\ dm \over \sqrt{1-l^2-m^2}}$$ (2.4.14)

This fundamental relationship between the visibility and the source intensity distribution is the basis of radio interferometry. In the optical literature this relationship is also referred to as the van Cittert-Zernike theorum.

Equation 2.4.15 resembles a Fourier transform. There are two situations in which it does reduce to a Fourier transform. The first is when the observations are confined to a the $U-V$ plane, i.e. when $w=0$. In this case we have

$$\mathcal{V}(u,v) = \int {\mathcal{I}(l,m) \over \sqrt{1-l^2-m^2}} e^{-i2\pi[lu+mv]} {dl\ dm }$$ (2.4.15)

i.e. the visibility $\mathcal{V}(u,v)$ is the Fourier transform of the modified brightness distribution ${\mathcal{I}(l,m) \over \sqrt{1-l^2-m^2}}$. The second situation is when the source brightness distribution is limited to a small region of the sky. This is a good approximation for arrays of parabolic antennas because each antenna responds only to sources which lie within its primary beam (see Chapter 3). The primary beam is typically $< 1^o$, which is a very small area of sky. In this case $n=\sqrt{1-l^2-m^2} \simeq 1$. Equation 2.4.15 then becomes

$$\mathcal{V}(u,v,w) = e^{-i2\pi w}\int \mathcal{I}(l,m) e^{-i2\pi[lu+mv]} dl\ dm$$ (2.4.16)

or if we define a modified visibility $\widetilde{\mathcal{V}}(u,v)=\mathcal{V}(u,v,w) e^{i2\pi w}$ we have

$$\widetilde{\mathcal{V}}(u,v) = \int \mathcal{I}(l,m) e^{-i2\pi[lu+mv]} dl\ dm$$ (2.4.17)

Footnotes

... field^2.5

We assume here for the moment that the electric field is a scalar quantity. See Chapter 15 for the extension to vector fields.

... by^2.6

Where we have invoked Huygens principle. A more rigorous proof would use scalar diffraction theory.

... shown^2.7

see for example, Christiansen & Hogbom, ``Radio telescopes'', Cambridge University Press