Interferometry

An interferometer typically consists of a number of sub-apertures positioned in a plane. Movements of the sub-apertures are used to produce a larger synthesized aperture plane. The aim of an interferometer is to measure the cross-correlation between the electric field measured at different points in the synthesized aperture plane. Light is collected with two or more sub-apertures (typically telescopes or siderostats) and a wavelength range is selected using optical filters. The simplest form of cross-correlation for an interferometer with two sub-apertures (and the fundamental goal of a measurement with a PRIMA FSU) is the correlated flux $C$ from the star:

$$C=\psi_{u1}\psi_{u2}^{*}$$ (6)

where $\psi_{u1}$ and $\psi_{u2}$ are the complex electric field amplitudes from the two sub-apertures produced by the star, and $^{*}$ indicates the complex conjugate. In VLTI/PRIMA the wavefronts from the star are spatially filtered (see [5] for a discussion of spatial filtering). This means that the flux from a single point in the image plane of each sub-aperture is selected, in contrast to the optical field as a function of position in the aperture plane described by $\psi_{u} \left(\mathbf{r},t\right)$ in Equation 1. In reality $C$ varies with time due to the change in projected baseline as the Earth rotates. In this discussion I will ignore the effects of Earth rotation on the projected baseline, which is adequate when describing a short astronomical observation.

The Earth's atmosphere introduces rapid fluctuations in the optical path length from an astronomical source to the two apertures, as discussed in Section 2.1. For monochromatic observations these optical path fluctuations can be described in terms of phase rotations $\phi_{p1}\left(t\right)$ and $\phi_{p2}\left(t\right)$ to $\psi_{u1}$ and $\psi_{u2}$ respectively in Equation 6 (in a similar way to Equation 2 in the aperture plane description). These phase rotations result in fluctuations in the phase of the correlated flux $C$. Again the amplitude fluctuations can be described by $\chi_{p1}\left(t\right)$ and $\chi_{p2}\left(t\right)$. Our description of the atmospherically perturbed correlated flux $C'\left ( t \right )$ then looks like this:

$$C'\left ( t \right ) = C\left ( \chi_{p1} \left(t\right) e^{i\phi_{p1}\left(t\right) }\right )\left ( \chi_{p2} \left(t\right) e^{i\phi_{p2}\left(t\right) }\right )^{*}$$ (7)

$$= CA\left ( t \right )e^{i\theta\left ( t \right ) }$$ (8)

where $i=\sqrt{-1}$ and $t$ is the time. $A\left ( t \right )$ and $\theta\left ( t \right )$ describe the amplitude and phase fluctuation introduced into the (complex) correlated flux by the atmosphere, and are given by:

$$A\left ( t \right ) = \chi_{p1}\left(t\right)\chi_{p2}\left(t\right)$$ (9)

$$\theta\left ( t \right ) = \phi_{p1}\left(t\right)-\phi_{p2}\left(t\right)$$ (10)

The timescale over which the RMS change in the phase $\theta\left ( t \right )$ at any given point is $\sim1$ radian is called the coherence time $t_{c}$ of the interference fringes. Each measurement of $C'\left ( t \right )$ must be made within one coherence time in order to avoid the measurements being corrupted by phase fluctuations.

VLTI PRIMA can observe two different stars simultaneously. One PRIMA FSU measures the correlated flux $C_{PS}'\left ( t \right )$ from the Primary Star (PS), and the other the correlated flux $C_{SeS}'\left ( t \right )$ on the Secondary Star (SeS).

In PRIMA all path lengths are measured relative to the metrology system. Any fluctuations in the metrology path lengths can be removed from the stellar beams by rotating the phase of the complex visibility. The primary OPD calibration will be given by switching the two stars between the different FSUs while keeping the metrology beams at fixed locations. The metrology system retro-reflector (RR2 and RR3 in the StS) will only be free of differential path fluctuations if the metrology beams are kept at a stable position in the image plane at M10. If the beams wander in the image plane, they will reflect off different points on RR3 giving OPD changes. It is absolutely essential that all the optics from M10 to the FSU are designed to minimise the movement of the beams in the image plane. The metrology system will cancel out path length errors after M10 to first order if the stellar beams are correctly superimposed on the metrology beams at the M10 image plane. The optical components in the stellar beam before M9 must be used to accurately steer the stellar beams onto the locations of the metrology spots.