In this section, we will describe the main parameters of an AO system, in order to give an estimate of the AO systems' current performances and limitations.

Uncorrected and Corrected SF MTFs

Uncorrected and Corrected SF MTFs

Normalized separation [1/D] Normalized angular frequency

Fig. 8. The left part of the figure reports the corrected phase structure functions for the cases D/r0 = 30, 20,10 (2., 3., and 4. respectively), together with the uncorrected one (1.). The effect of the wavefront correction is clearly visible in the saturation of the structure functions. The right side reports the corresponding MTFs computed from (19), with the diffraction limited MTF as reference (0.)

Normalized separation [1/D] Normalized angular frequency

Fig. 8. The left part of the figure reports the corrected phase structure functions for the cases D/r0 = 30, 20,10 (2., 3., and 4. respectively), together with the uncorrected one (1.). The effect of the wavefront correction is clearly visible in the saturation of the structure functions. The right side reports the corresponding MTFs computed from (19), with the diffraction limited MTF as reference (0.)

In Fig. 1, the basic layout of an AO system is reported to clarify where the relevant parameters come into play. We start by looking at the picture from the deformable mirror, going in a counterclockwise direction.

The relevant parameter for the deformable mirror is the number of degrees of freedom, usually given by the number of actuators. The aim of this device is to introduce in the incoming wavefront an optical path difference equal and opposite to the perturbation induced by the atmosphere at a given time. The sum of the two perturbations will give as a result a plane wave. As mentioned, the spatial scale of the atmospheric perturbation is given by the r0 value. Hence, in order to have the proper resolution on the corrective device, we need to have an actuator grid with pitch smaller then r0. In this case, the total number of actuators can be estimated to be of the order of (D/r0)2. The second parameter considered is the time required to compute and apply the correction. As we find in Sect. 3.3, the correlation time of the atmosphere is given in the Taylor hypothesis as t ^ r0/v, where v is the considered wind velocity. To be able to apply an efficient correction, the cycle time of the system has to be smaller than t so that we can write tcicle < r0/v.

Let us consider now the spatial sampling required to properly measure the wavefront aberration. As stated previously, the characteristic length of the atmospheric perturbation is given by r0. Assuming this, the wavefront spatial sampling has to be of the order of r0. A commonly used wavefront sensor in AO system is the so-called Shack-Hartmann sensor [20] which uses a lenslet array to obtain the wavefront slopes in several patches of the telescope entrance pupil. A sketch of the sensor is reported in Fig. 9. This will help the reader to have a better understanding of the basic parameter involved in the following calculations. As shown in the above picture, the lenslet array is optically conjugated to the telescope exit pupil. The array has a number of lenslets of the order of (D/r0)2. Each of the spots produced by the lenslet array is focused on a CCD detector. The intensity pattern of each spot is recorded for any given wavefront measurement. The spot displacement with respect to the zero or to the unperturbed position is measured by computing the intensity of center of gravity

i=i i=i where Ii and Xi are the intensity and the position of the i-th pixel considered. Moreover it is easy to show in geometrical optics approximation that

where w(x, y) is the wavefront perturbation and, fsh is the Shack-Hartmann lenslet focal length. The overbar denotes an average of the first derivative over the considered sub-aperture. Thus, the center of gravity is proportional to the wavefront slope in the considered sub-aperture defined by a particular lenslet. The measured data are used to write a linear system of finite difference equations relating the wavefront slopes to the phase difference on the various sub-apertures [12]. This system is usually solved by least square methods. The error in the reconstructed wavefront depends on the total number of photons received by each sub-aperture. This photon number is given by

in order to have 100 photons, per sub-aperture, per integration time5. Assuming D/r0 = 10, texp = 0.005, we need an overall flux $ = 2e6 or a star of visual magnitude 10. This number shows that the reference star to be used for wavefront sensing is relatively bright and is not always available in the surroundings of the faint astronomical object we want to observe. Finally, we note that the flux dependance on the third power of r0 tells that the flux requirements became very strong when the correcting wavelengths got shorter and shorter. A great advantage is usually obtained by performing the wave-front sensing at optical wavelengths and the adaptive correction in the NIR (usually 1-5|m). Doing so, the wavefront sensor can use a visible CCD camera (much faster and less critical than the IR detector). The adaptive correction is done in the J, H, and K bands, where r0 is larger and the requirements for all the parameters like number of actuators, sub-apertures, and time response, are all relaxed with respect to the visible band. This strategy is allowed

5 We assume here that 100 photons per sub-aperture per integration time are enough to have a good SNR in the center of gravity measurement. It can be shown that the accuracy in this measurement is given by 5xc = 6/^/N, where 6 and N are the lenslet spot full width half maximum and the received number of photons.

by the achromaticity of the optical path, as demonstrated in Sect. 3.2. This means that the correction done by the adaptive mirror, driven by an optical wavefront sensor, is valid for all the wavelengths. An adaptive Optics system working in this configuration is called polychromatic AO system. Finally, it is important to stress that for a polychromatic system the r0 we referred to above estimating the number of actuators and the number of sub-apertures is computed at the correcting wavelength (in the NIR).

5.3 Reference Source and its Angular Distance from the Scientific Object

The main effect that limits the usefulness of an AO system as described up to now is the so-called angular anisoplanatism. The angular anisoplanatism issue was analyzed by Fried [13] and later by other authors. We have already mentioned that because the reference star has to be considerably bright, the scientific target is rarely used as a reference star. Instead, a closely positioned star of high enough magnitude is used. To see how close the reference star has to be, we consider Fig. 10. In the figure, the telescope entrance pupil and a single turbulent layer are represented. As clearly shown, the two rays crossing the same point of the entrance pupil and coming from the scientific and reference objects, respectively, do cross the turbulent layer in two different points. In the sketch (Fig. 10), the reference source is located at an angular distance 0 = r0/h from the on-axis scientific target. The phase perturbations relevant to our discussion are also shown in Fig. 10 . In particular, is

the phase perturbation on the considered layer at high h; $2 is the phase perturbation experienced by the scientific object placed on-axis, while $3 is the one experienced by the off-axis reference star. The DM is driven to introduce the correction following the phase peak of $3. The result of this correction is represented as $4. This error is called named it such angular anisoplanatism, after Fried [13]: Now we have found as a coherence distance of the atmospheric perturbation the Fried parameter r0. With this in mind, it is easy to realize that the maximum angle 0 allowed between the reference and the scientific object is given by

this particular value of 0 called isoplanatic angle and is usually indicated as 00. The effect of angular separation between reference and scientific target is illustrated in Fig. 23. Substituting real values for r0 and h, like 0.2 m and 10 km, we find 9 = 4 arcsec. This angle is quite small and usually does not allow us to find the 10 mag star we need. It is exactly this problem that limits the sky coverage of an adaptive optics system or the fraction of sky that can be observed efficiently.

The sky coverage (SC) is defined as the percentage of the sky that can be observed using the AO system. This sky fraction is given by: SC = n02 ■ /4n, where 00 is the isoplanatic angle, is the number of stars having magnitude m < m0. Finally m0 is the AO system limiting magnitude identified using (27). Computing the SC requires a model for the star density as a function of the celestial coordinates. We summarize here all the equations obtained in previous sections and used to quantify the actual sky coverage of an AO system. The formulae we identified are:

Combining these basic equations J. Beckers [2] compiled a table that we present in Fig. 11. In this table, the various AO system parameters are identified as a function of the values of r0, t0 , and 00. Figure lists the sky coverage of the AO system as a function of the correcting wavelength. The sky coverage in the K band is 14% but it goes down to 5% in J band. In the visible wavelength regime the situation gets dramatically worst. This is mainly due to the third power dependance of the needed reference flux from r0 which in

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