Extension to a finite bandpass

Sections 2.2.1 to 2.2.5 are only strictly applicable at a single wavelength (i.e. for a monochromatic observation). The PRIMA FSUs utilize one broad channel and two narrow channels which together cover the K band (from $\sim1.95\mbox{ $\mu$m}$ to $\sim2.45\mbox{ $\mu$m}$). Due to the refractive properties of the atmosphere, each wavelength within the K band follows a slightly different path through the atmosphere. Light of different wavelengths reaching the aperture plane will have passed through slightly different parts of the turbulent layers in the atmosphere. The optical phase offset $\phi$ at each wavelength depends on both the mean phase across each telescope aperture and on the corrugations in the phase across each of the telescope apertures. The phase rotation $\phi\left( \mathbf{r},t,\lambda \right )$ (cf. Equation 2) in the aperture plane induced by a high-altitude atmospheric layer, at a given wavelength $\lambda$ and at a given time $t$ and position $\mathbf{r}$ in the AT aperture are dependent on the fluctuation in delay induced by the atmospheric turbulence $\Delta z\left( \mathbf{r'}+\mathbf{\Delta r'}\left ( \lambda \right ), t,\lambda \right )$ as follows:

$$\phi\left( \mathbf{r},t,\lambda \right ) =\frac{\Delta z\lef... ...elta r'}\left ( \lambda \right ),t,\lambda \right )}{\lambda}$$ (16)

where $\mathbf{r'}$ is the projection of $\mathbf{r}$ along the line of sight towards the star up to the layer of turbulence, and $\mathbf{\Delta r'}\left ( \lambda \right )$ is the wavelength-dependent offset from this path induced by bulk atmospheric refraction (the time delay for light propagation can be ignored).

In order to make measurements with high S/N ratio, the gradient of phase with wavelength must be minimised in the PRIMA FSU spectral channels. The phase difference between the correlated fluxes in the spectral channels at the edges of the K band is minimised through adjustments of the VLTI delay lines (group delay tracking). Note that it is the phase of $C_{PS}C_{SeS}^{*}$ for the central spectral channel which will be used for astrometry (i.e. the phase difference between the fringes on the two stars in the central spectral channel). The group delay will not be used for astrometric measurements as it cannot be measured as accurately as the phase of $C_{PS}C_{SeS}^{*}$, and because existing models for the refractive index of air provide lower accuracy when converting the group delay into the separation of the stars.