2 NGSs brightness

The limiting magnitude for NGS-based adaptive optics system can be easily computed by both numerical and analytical formulations (Rigaut et al. 1997). Matching with experimental results is relatively satisfactory (Roddier 1998). In the LGS case the extension is straightforward only accepting conical anisoplanatism degradation and omitting the tip-tilt treatment. Few experimental results are available to confirm such results. To my knowledge there is no published result on noise propagation in tomographic wavefront sensing. It should be pointed out that in the modal formulation given by Ragazzoni et al. (1999) it would be easier to approach the problem in an analytical way because error-propagation in matrices involves straightforward matrices manipulation. A careful analysis and numerical confirmation, however, still requires a thorough discussion that is beyond the scope of this work.

At the current state of knowledge it is very hard to make a firm projection of the limiting magnitude for tomographic wavefront sensing. In this section I try to outline some considerations useful for giving a rough estimate of such limiting magnitude.

I assume that the following statements are true:

1.

The strongest perturbing layer is located at, or very close to, the ground level;

2.

The number ${\cal N}$ of relevant perturbing layers is equal to the number of sensed NGSs;

3.

No portion of the highest layer intersected by the scientific cylindrical beam is unsampled.

Let us suppose that, under some given conditions for the Fried parameter r₀, Greenwood frequency $f_{\rm G}$, overall quantum efficiency of the wavefront sensing system q and its spectral bandwidth $\Delta \lambda$ expressed in nm, a certain zonal error in the wavefront correction $\sigma_0$ is obtained when a star of magnitude V₀ is used to close a classical adaptive optics loop. For each coherence zone (sized r₀) of the collected wavefront and in a single coherence time (of the order of $1/f_{\rm G}$), the number of collected photons N₀ will be given by (Zombeck 1990):

$$N_0 \approx 10^8 \; 10^{-0.4V_0} r_0^2 \frac{\Delta \lambda q}{f_{\rm G}}\cdot$$ (1)

We suppose also that the wavefront perturbation coming from the ${\cal N}$ layers is representative of most of the total turbulence experienced by the starlight coming into the telescope (grouping of very close layers can improve the effectiveness of the technique).

Each of these layers is numbered from 1 (the lowest) to ${\cal N}$ (the highest) and will be interested by an average number of NGSs given by n_i. For instance $n_1={\cal N}$ using the condition 1 and $n_{\cal N} \ge 1$ using the condition 3. For each layer one can define a r_0i characteristic of the wavefront passing just through that portion of turbulence.

Supposing the use of a star of magnitude V₀ to sense directly just the i-th single layer a number of photons given by:

$$N = N_0 \left( \frac{r_{0i}}{r_0}\right)^2$$ (2)

for each spatial coherence zone could be used. Since the variance error is proportional to the inverse of the square root of the number of collected photons (in the Poissonian dominated error approximation that we suppose reached by some state-of-the-art wavefront sensor) it turns out that the wavefront would be sensed with a zonal error $\sigma_i$ given by:

$$\sigma_i = \sigma_0 \frac{r_0}{r_{0i}}\cdot$$ (3)

How do these errors couple together when a tomographic approach is used? A pessimistic approach would tell us that sensing of a layer intersected by n_i NGSs will be affected by the sum of the errors coming from these sources. On the other hand an optimistic approach will tell us the opposite: averaging over n_i NGSs a gain is obtained because the related errors tend to cancel out. These considerations can be written in a compact form using the following:

$$\left(\frac{\sigma}{\sigma_0}\right)^2 = \sum_{i=1}^N n_i^\alpha \left( \frac{r_0}{r_0i} \right)^2$$ (4)

where $\alpha=1$ represent the pessimistic case and $\alpha=-1$ the optimistic one. The summation of the errors is incoherent. The case for $\alpha= \pm 2$ represent the (unlikely, and not treated here) case of coherent superpositions of the errors. In Eq. (4) it has been implicitly assumed that the weights for each zonal sensing are of the order of the unity. Of course this is a rather unjustified assumption. At this stage one can only imagine that very different weights combine together to mimic the situation coarsely expressed by Eq. (4). On the other hand the variations due to the choice on $\alpha$ span less than one order of magnitude in the $\sigma/\sigma_0$ ratio.

In order to reach with the tomographic approach the same zonal error as the classical single guide star approach, a limiting magnitude for each of the ${\cal N}$ NGS (the condition 2 is used here) of:

$$V = V_0 + \Delta V = V_0 - 2.5 \log \left( \frac{\sigma}{\sigma_0} \right)^2$$ (5)

must be obtained (here $\Delta V$ represent the variation due to the approach used). Using r₀ = 0.25 m, $f_{\rm G}= 200$ Hz, q=0.5, $\Delta \lambda = 300$ nm and an error $\sigma_0=1$ radian of wavefront phase, we can determine the number of required photocounts $N_0\approx 29$, being (Kern et al. 1989):

$$N_0 = \frac{2\pi^2}{\sigma_0^2}$$ (6)

and, by inversion of Eq. (1) a limiting magnitude V₀ = 13.0 is obtained. In order to estimate Eq. (4) we use two of the models by Roggermann et al. (1995), namely the SLC-N and the HV-21, using ${\cal N} =4$, n₁=n₂=4, n₃=2 and n₄=1 and scaling the various r_0i in order to match the overall r₀=0.25 m and reported for convenience of the reader in Table 1. Equations (4) and (5) gives $\Delta V = -1.3$ (a brighter limiting magnitude) for the $\alpha=1$ pessimistic case and $\Delta V = +1.6$ (the option to look for fainter NGSs) for the $\alpha=-1$optimistic case in the SLC-N model. For the HV-21 model $\Delta V=-1.4$($\alpha=1$) and $\Delta V=+1.5$ ($\alpha=-1$).

Conservatively we assume in the following $\Delta V =\pm1.0$, that is V=12.0 or V=14.0 as extrema of limiting magnitudes for the NGSs to be used in the tomographic approach. While no attempt is made to use many more fainter NGSs to mimic fewer brighter NGSs (an approach that could widen the practicability of the proposed approach) the approximate and tentative nature of these calculations must be reiterated. Also, it should be pointed out that essentially no literature is available on possible techniques for efficient wavefront sensing and correction in the tomographic mode and there is no evidence that straightforward sensing with classical wavefront sensor of each star to be coupled together in a dedicated wavefront computer represent the ultimate achievable performance. Moreover I have not speculated on the possibility of using NGSs to directly retrieve tomographic information as suggested for instance by Ribak (1995).

  

Table 1: The Fried parameters for the four layers model used in the text along with the right hand side of Eq. (4)