4 Phasing and path difference equalisation

 The sharpness S was maximised by exploiting its sine response to phase variations (Muller & Buffington 1974). In polychromatic light, the algorithm is applied separately to the different wavelength components in the x, y, $\lambda$ data cube.

The set of pupil phases is thus obtainable at each wavelength, and the modulo $2 \pi$ ambiguity can be removed to find the true optical path deviations and correct them. Only in such conditions of ideal interferometer adjustment can one obtain white images directly at the combined focus, with perfect colour registration and including any spectral components too faint for being separately imageable with the monochromatic algorithm. On resolved and colour-dependant objects, ensuring zero optical differences is made more difficult by the a priori unknown position indexing among the monochromatic images. One needs to solve for the value of integer k in Eq. (3)  

$$\delta= (k + \epsilon) \lambda + (\alpha x + \beta y),$$ (3)

where $\epsilon= \phi/2\pi$ and $\lambda$ is the wavelength. The term $\alpha x + \beta y$, where $\alpha$, $\beta$ are the sub-pupil's position coordinates, is the path difference generated by the wavelength-dependant image position error x, y. For a non-resolved star the shift x, y is identical at every wavelength and the second term of Eq. (3) therefore vanishes. One may then apply the classical fractional excess algorithm, used in the field of interferometric metrology since Michelson's measurement of the standard metre, to calculate the optical path differences. It consists in calculating, at every wavelength, $\delta$ values corresponding to all possible values of the unknown integer k. A $\delta$ value re-appearing at all wavelengths is the correct value of the optical path error for the sub-pupil considered. We have used this type of method, rather than the 3-dimensional Fourier transform, and verified the expected performance. Although not expected to work on resolved, colour dependant, objects, it also succeeded with such objects in most cases (Figs. 4, 5).

It appears possible to extend the method for dealing also with the shift term in Eq. (3), as required for full adjustment on resolved, colour-dependant, sources. Because the x, y image shift is identical for all sub-pupils, at a given wavelength, the corresponding equations are coupled and appear solvable if enough wavelengths are used. Different values of x, y and again the integer k can be inserted in the series of equations obtained at all wavelengths. The candidate values of $\delta$ thus obtained may be organised as a 2-dimensional array of cubes, each being derived from a pair of vectors $x_{\lambda}$, $y_{\lambda}$, specifying a trial set of position shifts. The cube axes are divided in units of k, $\lambda$ and the sub-pupil number. The particular cube providing the correct set of optical path errors is identified from the presence of numbers repeating in all lines within each of its k, $\lambda$ planes.

No attempt has yet been made to apply this method, although it may also solve classical problems of source position indexing in radio interferometry.