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3. Conical anisokinetism error determination

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.

3.1. Relative tilts within portions of the same wavefront

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 tex2html_wrap_inline922 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 tex2html_wrap_inline926 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 tex2html_wrap_inline922. 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 tex2html_wrap_inline926 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

3.2. Tilt tomography and conical-free tilt determination

We assume that (for istance using any of the cited techniques) the absolute tilt of an LGS is sensed.

Let us define tex2html_wrap_inline952 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 tex2html_wrap_inline956, where Hi and tex2html_wrap_inline960 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 tex2html_wrap_inline964 such that:


Equating the right sides of Eq. (4 (click here)) and Eq. (5 (click here)) one obtains:


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 tex2html_wrap_inline926, 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:


where the terms tex2html_wrap_inline972 and tex2html_wrap_inline974 on the right side of Eq. (9 (click here)) are experimentally measured, and the effective layer height tex2html_wrap_inline964 can be estimated as decribed in Ragazzoni et al. (1997). Care is to be given in a practical implementation to the effects of the error propagation from the estimation of the tex2html_wrap_inline926 angles and the tex2html_wrap_inline964 parameter (this last figure, morevoer, can change with time). The proposed approach can be seen as an extension of the constraints imposed in the TF90 scheme. Their Eq. (13), in fact, imposes a dummy zero-tilt condition on the system to be solved to have the tomography of the turbulence layers. Subsituting such a constraint by the full set of Eq. (8) is an equivalent way to obtain the absolute tilt free from the conical anisokinetism error. As a further remark one should recall that for practical reasons one should limits the number of perturbing layer sensed by the TF90 technique and it is to be evaluated the impact of this approach to the overall tilt error budget. A detailed calculation and an estimation of the achievable Sthrel ratio is well beyond the limits of this paper and is not treated here.

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