In the TF90 approach the overall tilt term can be retrieved if a suitable NGS is available in the field. In the present approach we show in the following that the knowledge of the absolute tilt of only one of the various LGS allow to obtain the absolute tilt free from the conical anisokinetism effect.
Let us consider a wavefront distortion W(x,y) introduced by a turbulence layer and defined in some domain of the (x,y) plane at the height of the layer itself (see Fig. 1 (click here) for an unidimensional section). Let us also consider two sub-domains of this plane denoted by the indexes A and B. While the absolute tilt of W can be unknown, the knowledge of the higher order shape of the wavefront enables us to define uniquely the relative tilt measured considering the wavefront portions A and B. Such a quantity is denoted in the following by and does not depends upon the absolute tilt of W. For each of the layers in the TF90 technique one can figure out any quantity of this type between single portions of the same wavefront. As a particular case (see Fig. 2 (click here)) TF90 allows for the determination of the detailed wavefront shape for each layer as seen on the cylindrical shape beam going from the telescope entrance to the astronomical target. For any of the projected LGSs forming the constellation required to the proper turbulence tomography one can define as the relative tilt between the tilt generated by the i-th layer for the astronomical target and the tilt originated in the sub-domain defined by that LGS cone.
Figure 1: In this unidimensional section of the wavefront W(x,y) the relative tilt difference experienced by the wavefront in the domains A and B is denoted by . This quantity does not depends upon the absoulte tilt knowledge of W in the whole domain
Figure 2: For any i-th layer the quantity is defined as the differential tilt produced by that layer into the two domains A related to an infinitely distant astronomical target and B related to a particular LGS used for the TF90 tomographyc sensing
We assume that (for istance using any of the cited techniques) the absolute tilt of an LGS is sensed.
Let us define as the angular tilt introduced by the sub-domain of the i-th layer intersected by the LGS beam. Because of the conical shape of the LGS beam, the optical path disturbance propagates from the layer to the telescope aperture with the same magnitude, but it is distributed on a larger surface. This effect reduces the angular LGS tilt contribution by a factor , where Hi and are the heights of the layer and of the LGS respectively. This factor is given by the ratio between the diameter of the cone at the height Hi and at the telescope level. The measured angular tilt of the LGS is given by:
Following Ragazzoni et al. (1997), with the use of Eq. (4 (click here)) we define an effective tilting layer height such that:
On the other hand one can write the similar subdivision over the whole set of k layers for the cylindrical shaped beam of an hypotethical Natural Guide Star (NGS):
The complete set of , that are estimated through the TF90 technique, are given by:
Summing over the whole k layers the latter relationship, and using the Eq. (5 (click here)), one can easily derive the following: