Spatial coherence function

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An astrophysical phenomenon occurs at location R. This phenomenon causes a time-variable electric field E(R,t).

Maxwell's equations say: an electromagnetic wave will propagate away from that point. An astronomer observes the wave at a position r.

If we have a varying field with finite time interval, we can express the magnitude of the field as the real part of the sum of the Fourier series. In Fourier series, only time varying functions are simple exponentials.

For astrophysical phenomenon Maxwell's equation are linear. So, we can deal with the coefficients of this Fourier series instead of the time varying electric field. Coefficients of this Fourier series are called quasi-monochromatic components which are complex quantities. As Maxwell's equations are linear, we can superpose the fields by various source points,

<math>\displaystyle \mathbf{E}_\nu(\mathbf{r}) = \int\int\int P_\nu(\mathbf{R},\mathbf{r})\mathbf{E}_\nu(\mathbf{R})dxdydz</math>

The function P is the propagator that describes how the field at R influences the field at r.

Assumptions

Now let's get on with the simplifying assumptions. There are 5 to be exact:

  1. The electromagnetic radiation is a scalar field. (ignore all polarization phenomena)
  2. The sources are a long way away from us.
  3. The space within the celestial sphere (imaginary of radius R) is empty.
  4. Radiation from astronomical objects is not spatially coherent.
  5. It has two forms:
    • The vectors <math>r_1-r_2</math> lie in a plane (measurements done in a plane (u,v) or favored coordinates)
    • The endpoints of the vectors s lie in a plane (all radiation comes from a small portion of the celestial sphere)

Now let's see the consequences of these assumptions:

Assumption 01

First assumption enables the multiplication in the previous equation to be regarded as ordinary scalar multiplication and also the propagator becomes a scalar function.

Assumption 02

Second assumption forces us to give up all hope of measuring the radiation in the third dimension, we can than only measure the //surface brightness//. Thus we can conceive a celestial sphere between the source and us where the field strength E will be measured. So we learn about the distribution of the source of the field by measuring the electric field on the surface of the celestial sphere.

Assumption 03

Third assumption according to Huygens' principle makes the propagator a very simple function,

<math>E_\nu(\mathbf{r}) = \int \mathcal{E}_\nu(\mathbf{R}) \frac{e^{2\pi i\nu|\mathbf{R}-\mathbf{r}|/c}}{|\mathbf{R}-\mathbf{r}|} dS</math>
Assumption 04

The correlation of the field at two different locations is defined as the expectation of a product,

<math>V_\nu(\mathbf{r_1},\mathbf{r_2}) = \left\langle \mathbf{E}_\nu(\mathbf{r_1})\mathbf{E}^*_\nu(\mathbf{r_2}) \right\rangle</math>

Now we can put the values of the field and its complex conjugate in this equation to get,

<math>V_{\nu}(\mathbf{r}_1,\mathbf{r}_2) = \left\langle \int\int \mathcal{E}_{\nu}(\mathbf{R}_1) \mathcal{E}^*_{\nu}(\mathbf{R}_2) \frac{e^{2\pi i\nu |\mathbf{R}_1-\mathbf{r}_1|/c}}{|\mathbf{R}_1-\mathbf{r}_1|} \frac{e^{-2\pi i\nu |\mathbf{R}_2-\mathbf{r}_2|/c}}{|\mathbf{R}_2-\mathbf{r}_2|}~ dS_1 dS_2 \right \rangle</math>

Fourth assumption means,

<math>\left\langle \mathcal{E}_\nu(\mathbf{R}_1)\mathcal{E}_\nu(\mathbf{R}_2) \right\rangle = 0, \mathbf{R_1} \ne \mathbf{R}_2</math>

Now let's put,

<math>\hat{s} = \frac{\mathbf{R}}{|\mathbf{R}|}, I_\nu(\hat{s}) = |\mathbf{R}|^2 \left\langle |\mathcal{E}_\nu(\hat{s})|^2 \right\rangle, dS = |\mathbf{R}|^2 d\Omega</math>

Then, using the second assumption neglect the small terms of order <math>|r/R|</math> and replace dS with that mentioned above to get,

<math>V_\nu(\mathbf{r}_1,\mathbf{r}_2) \approx \int I_\nu(\hat{s}) e^{-2\pi i\nu s.(\mathbf{r}_1-\mathbf{r}_2)/c} d\Omega</math>

This is called the spatial coherence function.

Assumption 05

First form of the fifth assumption enables us to make the measurements in a favored coordinate system by writing the vector spacing of the separation variable in the coherence function as,

<math>\mathbf{r}_1-\mathbf{r}_2 = \lambda(u,v,0)</math>

Than the components of the unit vector are,

<math>\hat{s} = (l,m,\sqrt{1-l^2-m^2})</math>

In the new coordinate system,

<math>V_\nu(u,v,w\equiv 0) = \int\int I_\nu(l,m) \frac{e^{-2\pi i(ul+vm)}}{\sqrt{1-l^2-m^2}} dl dm</math>

This is a Fourier transform relation between the spatial coherence function and the modified intensity.

Second form of the fifth assumption implies,

<math>\hat{s} = \hat{s}_0 + \sigma</math>
<math>1 = |\hat{s}| = \hat{s}.\hat{s} = \hat{s}_0 . \hat{s}_0 = \hat{s}_0 . \hat{s}_0 +2\hat{s}_0 . \sigma + \sigma.\sigma \approx 1+2\hat{s}_0.\sigma</math>

Again imply a special coordinate system such that,

<math>\hat{s}_0 = (0,0,1)</math>

Thus we get a different form of the spatial coherence function,

<math>V'_\nu(u,v,w) = e^{-2\pi iw} \int\int I_\nu(l,m) e^{-2\pi i(ul+vm)} dl dm</math>
<math>\Rightarrow V_\nu(u,v) = e^{2\pi iw} V'_\nu(u,v,w) = \int\int I_\nu(l,m) e^{-2\pi i(ul+vm)} dl dm</math>

This is the coherence function relative to the direction s_0 which is called the phase tracking center. As this equation is a Fourier transform we can write the direct inversion as,

<math>I_\nu(l,m) = \int\int V_\nu(u,v) e^{2\pi i(ul+vm)} du dv</math>


The relationship between the two forms of the fifth assumption can be seen in the light of the systematic role played by two vectors s and <math>r_1-r_2</math> in the main equation of the spatial coherence function.

Discrete sampling

The spatial coherence function is not known everywhere in practice but sampled at particular places on the u-v plane. Sampling si described by a sampling function, S(u,v) which is zero when no data have been taken. Than one can calculate a function,

<math>I^D_\nu(l,m) = \int\int V_\nu(u,v) S(u,v) e^{2\pi i(ul+vm)} du dv</math>

This is called the dirty image. Relation between the dirty image and the desired intensity is,

<math>I^D_\nu = I_\nu * B</math>
<math>B(l,m) = \int\int S(u,v) e^{2\pi i(ul+vm)} du dv</math>

Where B is the synthesized beam or the point spread function (PSF) corresponding to the sampling function.

Element reception pattern

The interferometer elements are not point probes which sense the voltage at that point, but are elements of finite size, which have sensitivity to the direction of arrival of the radio radiation. This sensitivity as a function of direction is described by primary beam or normalized reception pattern of the interferometer elements. Coherence function including this factor becomes the so called complex visibility,

<math>V_\nu(u,v) = \int\int \mathcal{A}_\nu(l,m) I_\nu(l,m) e^{-2\pi i(ul+vm)} dl dm</math>

This is relative to the chosen phase tracking center. Thus the second form of the fifth assumption is very useful in case of direction sensitivity. Primary beam falls rapidly to zero except in the vicinity of some phase tracking centers, i.e. the pointing center for the array elements.[১]

References

  1. B. G. Clark, Coherence in Radio Astronomy, NRAO, Socorro, New Mexico, USA