In this section it is assumed that only a single, thin layer, is responsible
for the star wandering. We recall here that the multicolour LGS absolute
tilt determination technique is based on the
measurement of the differential angle of arrival of two different colour LGSs
excited by a monochromatic laser beam.
Following Foy et al. (1995), when the layer is introducing a tilt
at a given primary wavelenght, the
two colour beams will experience a deflection's difference 
given by:
where n is the refractive index at the primary wavelength and 
is the variation of refractive index of the air between the two LGS colours.
The tilting layer is placed at height h while the LGS is excited at
height H. While the tilting layer can take place in any h = 0
20 km, the
mesospheric Sodium LGS exhibits a characteristic height H = 95 km (Happer
et al. 1994).
In the following we refer to RED and BLUE respectively to the
longest and shortest wavelengths of the two LGS colours. Moreover, we'll assume
that the exciting beam is a red one, and that the LGS will produce both a
red and a blue beam; for sake of simplicity we also assume that
the laser firing and the AO observations are performed at the local zenith.
None of these assumptions will affect significantly the conclusions.
Referring to Fig. 1 (click here), we consider the chief ray of the beam from the LGS to the observing telescope, where the pupil is located.
Figure 1: The apparent displacement 
between two colour artificial beacons
as seen by the ground observer is slightly different than the quantity
. Their ratio depends upon the h altitude. It is also shown in
the upper-left corner inset how, assumming 
, the
measured angle 
strictly coincides with 
The 
position of the LGS following the two colour beams can be easily
derived under the assumption that all the considered angles are very small.
For the red beam:
while, following the blue laser beam path:
Equating the expressions given in Eqs. (2) and (3) we get:
From Eq. (4) one can see that the measurement 
will depend upon both
and h. In the case of a single layer with a constant altitude this
will translate into the introduction of a scale factor close to 1. In
fact, h is usually less than 
. However, a time evolving value
for h will introduce an apparent relative displacement 
of the red
and blue beacons undistinguishable from the displacement introduced by the
temporal evolution of the tilt 
.
One can expect that, using the more realistic multi-layer approach,
some spreading of the effective h can introduce an error source in the
absolute tilt determination.