Two important assumptions have to be made before we go any further. Those conditions are not necessarily fully satisfied, but they are required if one wants to develop an analytical expression of the interferogram.
In a modal description of atmospheric turbulence, the piston corresponds to the most fundamental perturbation, i.e. the fluctuations of the average phase of the corrugated wavefront. The piston mode on a single pupil does not modify the state of coherence of that pupil and therefore, does affect neither the image quality nor the coupling efficiency into a single-mode fiber. The differential piston between two independent pupils, however, is equivalent to the addition of a small random delay in the OPD.
Recently, Perrin (1997) proposed a numerical method to remove the differential piston after data acquisition. The piston can also be eliminated in the instrument before data acquisition if the pupils are cophased with a fringe tracker, as it is the case for example in the Mark III interferometer (Shao et al. 1988). A fringe tracker based on guided optics has already been proposed (Rohloff & Leinert 1991). Cophasing the pupils offers the additional advantage to considerably improve the sensitivity because integration times can in principle be arbitrarily long (Mariotti 1993), but it requires to build a dedicated active system.
Without piston, the OPD variation is uniform. The signals are recorded as
time sequences, but if we assume that the fringe speed v is constant
during a scan there is a direct linear relationship between the time
variable t and the position variable (the global OPD) x. The signals
are sampled at equal time intervals 
, which correspond to equal
length intervals 
.
When taking the Fourier transform of the signals, the conjugate variable
with respect to position x is the wave number 
, and the frequency
is the conjugate variable for the temporal sequences. It is
important to keep this duality in mind in order to be able to reason
alternatively in terms of time/frequency or position/wave number. Actually,
in this paper either one or the other variable pair is used depending on
the needs, and the variable change that it sometimes implies shall be
implicit.
Usually the starlight injection efficiency 
is both a function of
time t and of wave number 
, since the structures of both
E
and E
depend on wavelength, and E
is
determined by the instantaneous state of the atmospheric turbulence. For
what follows it is necessary to assume that the time and wave number
variables can be separated in the coupling efficiency coefficient, so that
we can write
A heuristic justification is given here. For a diffraction limited image,
the electric field morphology at the focus of the telescope depends on
wavelength 
only through a scaling factor. Within the practical
range of optical frequencies at which a single-mode fiber can be operated,
the fundamental mode can be approximated by a Gaussian function whose width
is proportional to 
(Neumann 1988). Thus the two
fields change homothetically with respect to wavelength, and the overlap
integral (Eq. 3 (click here)) remains almost unchanged. It follows
that the injection efficiency is quasi achromatic for a diffraction limited
image.
Things are different for a stellar source, but if we assume that the
turbulence is weak (
, where d is the diameter of the
pupil and 
the Fried parameter (Fried 1966)), tip-tilt
modes dominate the atmospheric turbulence (Noll 1976) and the
image of the star can be modeled by a unique speckle randomly walking
around its nominal position. The speckle offset with respect to the fiber
core is a function of time exclusively, whereas the sensitivity of 
to that offset depends on the color only. It is thus reasonable to assume
that the time and wave number variables can be separated.