AO systems were developed from the need to compensate for atmospheric distortions, i.e., reducing the image degradation due to turbulence. A wavefront sensor (WFS) measures incoming wavefronts from a guide star (GS). The real-time control (RTC) unit computes from these measurements the optimal correcting shape of adeformable mirror (DM). The reflection on the DM is compensating then for the atmospheric tur-bulence. The update of the DM shape needs to be done in less than a millisecond, thus the measurements of the WFS and the calculations of the RTC have to be performed at the same frequency.
First ideas for AO systems were developed in [6]. A sketch of an AO system is presented in Figure 2.7. The beamsplitter is necessary as not all incoming light should go to the WFS, but the major part should be left to observe a corrected high resolution image with the scientific camera.
2.3.1 Guide star
Aguide star (GS) is a bright astronomical object, e.g., a star, which is used for getting information on the turbulence in the atmosphere. The measurements from this guide star are the input for the reconstruction process of the RTC. Guide stars can be ei-ther natural astronomical objects that are point sources, so-callednatural guide stars (NGS), or artificial ones, so-called laser guide stars (LGS). The reason for creating artificial guide stars is the limited sky-coverage with NGS.
An LGS is formed by shooting a strong laser up into the night sky, which stimulates a sodium layer at approximately 90 km height in the atmosphere and thus creates an artificial astronomical object.
Effects of LGS
From the creation of LGS by stimulation of a sodium layer three major effects arise that need to be taken into account: Spot elongation, cone effect and tip-tilt indetermination.
20 CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS
Figure 2.7: A general AO system, source [29].
The thickness of this sodium layer is modeled by a Gaussian random variable via its mean hLGS and a full width half maximum parameter FWHMN a. Usually, in models hLGS “90kmand FWHMN a“11.4kmis assumed. Note that FWHM for a Gaussian random variable relates to the standard deviationσ as
FWHM“2?
2 ln 2¨σ.
Due to the “thickness” of the sodium layer, its vertical width, the scattering of the laser beam is not a single point as for an NGS, but rather a small stripe on the night sky.
The spot, registered by a detector on the telescope, appears elongated, see Figure 2.8.
This effect is calledspot elongation. This elongation degrades the measurement accu-racy and the error increases linearly with the elongation of the spot in the direction of the centroid [23]. Furthermore, spot elongation introduces correlation between the X and Y measurements in the subaperture [161]. We follow the lines of [23], when discussing the compensation of spot elongation in Chapter 5.
Additionally to the vertical width of the sodium layer, its finite height causes the light to travel through a cone-like volume in the atmosphere. This effect is referred to as
CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS 21
Figure 2.8: Graphical representation of the photon distribution of an NGS (left) and the spot elongation generated by an LGS (right) on the detector, from [178].
cone effect, illustrated in Figure 2.9. In atmospheric tomography, as in Section 7.2, the cone effect has to be taken into account as a scaling factor cl relating hLGS and the height of an atmospheric layerhl.
Figure 2.9: Illustration of the cone effect of an LGS, from [178].
A third complication arising from LGS is the so-calledtip-tilt indetermination, meaning that the planar part of the incoming wavefront is wrongly observed at the telescope.
This effect stems from the fact that the laser beam passes through the same atmosphere
22 CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS
twice, a first time when traveling up and a second one when being back scattered from the sodium layer. Thus, a tip or tilt of the incoming wavefront due to, e.g., a layer of differently tempered air, as shown in Figure 2.10, cannot be observed and the “correct”
position of the LGS remains unknown. As remedy for this phenomenon additional NGS are used, from which just the tip-tilt information is obtained, as their exact positions are known. Detailed discussions on this effect and how it can be overcome can be found, e.g., in [149, 178, 165].
Figure 2.10: Illustration of the tip-tilt indetermination, from [3].
2.3.2 Wavefront sensor
The device to measure light coming from a guide star is calledwavefront sensor (WFS).
A high enough spatial resolution and measurement speed are important requirements for a successful real time compensation by the AO system. The techniques developed vary from focal plane techniques to pupil plane techniques [138]. Most current WFS, such as the Shack-Hartmann WFS and Pyramid WFS (cf, e.g., [129, 131, 130]), provide indirect measurements of the atmospheric distortions. These are two pupil plane WFS, which are commonly used nowadays. Apart from these also other sensors, such as the curvature WFS (cf [137]), exist.
Shack-Hartmann WFS
The Shack-Hartmann WFS (SH-WFS) [40, 152, 123] consists of an array of small lense-lets and a photo detector lying behind. These sensors measure the average gradient of the incoming wavefront in each subaperture. More specifically, the detector mea-sures the x- and y-coordinates of the points where the light of each lenslet is focused.
These measurements are related to the slope of the incoming wavefront via the center
CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS 23
of gravity of the focal spots. To compute the center of gravity, a variety of methods exists, such as (weighted-)centroiding or matched filtering, e.g., [61]. The principle of a SH-WFS is illustrated in Figure 2.11.
Usually having one Shack-Hartmann measurement over the whole aperture is not enough, so Shack-Hartmann WFS consist of a nˆn grid of apertures as described above, now called subapertures. Such a sensor covers the whole aperture of the tele-scope and Ω“ Yni,j“1Ωij, where Ω is the telescope aperture and Ωij is one aperture of the SH-WFS. Not all subapertures are illuminated if the sensor is square. One thus has to distinguish between active and inactive subapertures.
The action of one subaperture Ωij of the SH-WFS on the incoming wavefront can be formalized by the operator Γ :Hs ÑR2n
2 as sxri, js “ pΓxϕqri, js:“ 1
|Ωij| ż
Ωij
Bϕ
Bxpx, yqdpx, yq, syri, js “ pΓyϕqri, js:“ 1
|Ωij| ż
Ωij
Bϕ
Bypx, yqdpx, yq, fori, j P t1, . . . , nu.
Note that Γ is well-defined forsą 12 according to [116]. The waffle mode (checkerboard pattern) and the piston mode (constant function) lie in the nullspaceNpΓq. However, the piston mode can be neglected for the wavefront reconstruction. The additional condition
ż
Ω
ϕpx, yqdpx, yq “ 0 (2.20) enforces a piston mode zero forϕon Ω in the reconstruction process.
We combine the operators Γx,Γy to a system of equations on the whole telescope aperture, defined by
Γϕ“s, Γ“ pΓx,Γyq, s“ psx, syq. (2.21) In a discretized version, the action of Γ on ϕcan be rewritten as
sxri, js “ pϕri, j`1s ´ϕri, jsq ` pϕri`1, j`1s ´ϕri`1, jsq
2 ,
sxri, js “ pϕri`1, js ´ϕri, jsq ` pϕri`1, j`1s ´ϕri, j `1sq
2 ,
whereϕri, js denotes the wavefront evaluated in the subaperture with index pairpi, jq [149].
24 CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS
Figure 2.11: Principle of a Shack-Hartmann WFS, source [163].
2.3.3 Deformable mirror
The deformable mirror (DM) is a thin, flexible and highly reflective mirror that can be moved and its shape can be controlled by actuators attached from below, as il-lustrated in Figure 2.12. While in many current telescopes the DM consists of a continuous faceplate, the DM of the ELT will be different as it will consist of several hexagonal segments. Around 5200 actuators are planned to allow the surface to be readjusted in less than a millisecond. Each segment will be made of ceramic glass at a thickness of approximately 2 millimeters. The actual size of the DM is around 2.5 meters in diameter, while still optically scaled to the aperture size of the telescope [167, 18]. The actuators are controlled by applying voltages.
2.3.4 Real-time control
The shape of the DM, and moreover the commands for the actuators of the DM, are derived from the WFS measurements by the real-time control (RTC) unit. For this action specific fast algorithms are required. We will develop algorithms for different AO systems in Chapter 4 and 5.