Closure phase and the Poincaré sphere

In this section we use right- and left-circular polarization states as the basis. A general elliptically polarized state can be written as a superposition of two states represented by the vector $$\left( \begin{array}{ll} \cos \theta/2 \sin \theta/2 e^{\iota \phi} \\ \end{array} \right)$$. Clearly, $\theta = \pi/2$ corresponds to linear polarization and $\theta \ne 0,\pi/2$ to elliptical polarization (see Fig. 7.4). Increasing $\phi$ by $\zeta$ rotates the direction of the linear state or the major axis of the ellipse by $\zeta/2$. We can chose the phase of the basis so that $\phi=0$ corresponding to linear polarization along the x-axis. The Poincaré sphere representation of the state of polarization maps the general elliptic state to the point ( $\theta, \phi$) on the sphere. The properties of this representation are reviewed by Ramachandran & Ramaseshan (1961). We are concerned here with one remarkable property, discovered by Pancharatnam (1956); Pancharatnam (1975). Whenever there is constructive interference between two sources of radiation, it is natural to regard them as in phase. A remarkable property of this simple definition manifests itself when we consider 3 sources of radiation of different polarization - that if a source A is in phase with B and B in phase with C, C in general need not be in phase with A. The phase difference between A and C is known in the optics literature as the geometric or Pancharatanam phase (see also Ramaseshan & Nityananda (1986); Berry (1987)). We show that this naturally occurs in radio interferometry of an unpolarized source with three antennas of different polarizations.

Figure 7.4: Poincaré sphere representation of the state of polarization. Poitns on north and south poles of the sphere correspond to pure right and left circular polarization. Points along the equator correspond to pure linear polarization. A linear polarization along the x-axis is represented by a point at $\theta = \pi/2$ and $\phi=0$ while linear polarization along the y-axis is represented by a point at $\theta = \pi/2$ and $\phi=\pi/2$.

Let the polarization states of the three antennas be represented by $\left( \begin{array}{ll} g_1 \alpha_1 \end{array} \right)$, $\left( \begin{array}{ll} g_2 \alpha_2 \end{array} \right)$, and $\left( \begin{array}{ll} g_3 \alpha_3 \end{array} \right)$ in a circular basis. Denoting the vector $\left(\begin{array}{ll} g_i \\ \alpha_i \end{array} \right)$ by $\psi_i$, one clearly see that the visibility on the 1-2 baseline is proportional to $\psi_1^\dag\psi_2$. Hence the closure phase around a triangle made by antennas 1, 2, and 3 is the phase of the complex number (also called the triple product) $V_{123}=(\psi^\dag _1 \psi_2)(\psi^\dag _2 \psi_3)(\psi^\dag _3 \psi_1)$. In the quantum mechanical literature, this type of quantity goes by the name of Bargmann's invariant and its connection to the geometric phase was made clear by Samuel & Bhandari (1988). With some work, one can give a general proof that the closure phase (phase of $V_{123}$) is equal to half the solid angle subtended at the centre of the Poincaré sphere by the points represented by $\psi_1$, $\psi_2$, and $\psi_3$ on the surface of the sphere. For the case where the polarization state of the three antennas are same, this phase is zero in general. However, when the polarization states of the antennas are different, this phase is non-zero.

The well known result that an arbitrary polarization state can be represented as a superposition of two orthogonal polarization states translates to representing any point on the Poincaré sphere by the superposition of two diametrically opposite states on a great circle passing through that point. For example, circular polarization can be expressed by two linear polarizations, each with intensity $1/\sqrt{2}$. In the context of the present work, the nominally circularly polarized antenna maps to a point away from the equator on the Poincaré sphere (it would be exactly on the pole if it is purely circular) while the rest of the antennas map close to the equator (they would be exactly on the equator if they are purely linear and map to a single point if they were also identical). The visibility phase due to the extra baseline based term in Equation 7.3 due to polarization mis-match is a consequence of the Pancharatanam phase mentioned above. This phase, on a triangle involving the circularly polarized antenna, will be close to the angle between the two linear antennas. For example, if $\psi_1=\left( \begin{array}{ll} 1 \iota \end{array} \right)$, $\psi_2=\left( \begin{array}{ll} 1 0 \\ \end{array} \right)$, and $\psi_3=\left(\begin{array}{ll} \cos \gamma \sin \gamma \end{array} \right)$, the phase of $V_{123}$ will be $\gamma$. This picture can be depicted by plotting the real and imaginary parts of ${\alpha_i^{\it q}}/{g_i^{\it p}}$, which is done in Fig. 7.3. The circularly polarized antenna can be clearly located in this figure as the set of point away from the origin while the linearly polarized antennas as the set of points close to the origin. The collection of points located away but almost symmetrically about the origin represents the nominal right- and left-circularly polarized feeds. Points on the equator, but significantly away from the origin represents an imperfect linearly polarized antenna. Note that the average closure phase between the nominally linear antennas is close to zero, which defines the mean reference frame in Fig. 7.3.