Jones matrix
The observed visibilities of an interferometer and the actual sky brightness matrix can be related through Jones matrices. They are a collection of linear transformation matrices that represent all the effects along the propagation path of the desired signal including the astronomical, atmospheric and instrumental effects. It can be seen as <math>\mathbf{J}</math> in the RIME,
- <math>V_{pq} = \mathbf{J}_p B \mathbf{J}_q^H</math>
Some Jones matrices
| Characteristics | Scalar | Diagonal | Non-zero off-diagonal |
|---|---|---|---|
| Commute | with everything | with themselves | with themselves |
| Coordinate | independent | dependent | dependent |
| Effect on <math>\mathbf{e}</math> | affects both components equally | affects the components independently | intermix the two components |
| Example | phase delay matrix | gain of the antenna | rotation matrix |
Phase delay matrix
Correlator introduces additional delay terms to compensate for the pathlength difference in the chosen direction, the chosen direction being the phase centre. Let's say antenna p is located at the point <math>(u_p,v_p,w_p)</math>. The phase difference at point <math>u_p</math> w.r.t.the point <math>u=0</math> is for a signal coming from <math>\sigma</math> direction is,
- <math>k_p = 2\pi\lambda^{-1} [u_pl+v_pm+w_p(n-1)]</math>
where <math>l,m,n=\sqrt{1-l^2-m^2}</math> are the direction cosines of <math>\sigma</math>. The <math>\lambda</math> term can be avoided if we express u in terms of wavelengths. Thus we can introduce scalar Jones matrices,
- <math>K_p = e^{-ik_p} = e^{-2\pi i[u_pl+v_pm+w_p(n-1)]}</math>
- <math>K_q = e^{-ik_q} = e^{-2\pi i[u_ql+v_qm+w_q(n-1)]}</math>
If there is no contamination other than the phase delay the visibility matrix is,
- <math>V_{pq} = K_p B K_q^H = B e^{-2\pi i[u_{pq}l+v_{pq}m+w_{pq}(n-1)]} = X_{pq}</math>
where, <math>u_p-u_q=u_{pq}</math>.
This visibility matrix is called the source coherency and written as <math>X_{pq}</math>. We can say that the visibility of a single point source if there is no contamination other than phase delay is called the coherency.
But actually there are some other contamination due to the antenna gain (G) and thus,
- <math>V_{pq} = G_p X_{pq} G_q^H</math>
