Measures in AO systems

2.6.1 Quantities and sizes in AO systems

Within AO systems certain system specific quantities appear that are related either to the observation or that are inherited by the technical set up. We discuss the quantities that are needed throughout this work.

Minute and second of arc

Positions of stars on the sky are often given in angular coordinates and also we will often refer to arcmin (1’) and arcsec (1”). Both of them are angular measurements, being 1{21600 and 1{1296000 of a circle, respectively, and thus given by

1¹ “ π

10800rad„2.9¨10^´4, 1² “ π

648000rad „4.8581¨10^´6.

CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS 29

Field of view (FoV)

The field of view (FoV) is usually described by the diameter of a circle in arcmin or arcsec. Two different FoV have to be distinquished: the science FoV and corrected FoV. The first one can be much smaller than the latter one, especially for complex AO systems. We will most of the time refer to the latter one which is determined by the guide star asterism.

Frame rate

The duration of sensing at the CCD of the WFS is called theframe rate of a specific AO system. For many AO system, it lies around 500 Hz, meaning that the time frame for calculating the DM shape(s) is approximately one millisecond. By the frame rate, the length of one time step in simulations is determined.

Wavelength λ

Observations can be performed in different wavelengths and, as already discussed, many parameters are wavelength dependent. In this work, we focus on observations mostly inK-band, i.e., λP r2.0,2.4sµm, being a near infrared wavelength. One has to distinguish between sensing and evaluation wavelength, which do not need to coincide.

In general, the AO performance can change drastically with changingλ.

Photon flux n_ph

The intensity of light reaching the aperture (and consequently the WFS) is measured in photons. Typically the photon flux denotes the number of photons that reach one subaperture of the WFS per frame. In our tests, the photon flux ranges from low (1 to 500 photons per subaperture per frame) to high (10000 and more) flux.

The photon flux is directly related to themagnitude m, the astronomical measure for brightness of an object, at a certain wavelengthλ, through several formulae described in [156], which are recalled in the following. We denote withF₀ the flux in Jy (Jansky) (i.e., [10^´26W m^´2Hz^´1s) at magnitude zero for a given wavelength, being a fixed quantity known from measurements. F_λ is the spectral flux density at wavelength λ at magnitude m and related to F₀ by

F_λ “10^´0.4¨m¨F₀¨β{λ²,

whereβ is a constant to convert between different units, in our case β “3¨10^´12. The photon flux F_ph is then computed as

F_ph“F_λ{E_ph¨∆λ,

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with the photon engery E_ph “h¨c{λ, ∆λ being the width of the wavelength band, h the Planck constant and cthe speed of light.

Finally, n_ph can be calculated as

n_ph“F_ph¨QE_R¨T_atm¨T_optics¨10^´3 ¨∆t,

whereQE_Ris the number of electrons per photon,T_atmandT_opticsare the transmission factors of the atmosphere and the telescope optics, respectively, and ∆t is the time per frame, calculated from the frame rate. Note that some of the used quantities are constant, while others depend on the used observing system and wavelength.

2.6.2 Quality measures

In order to evaluate the quality of the run of an AO system, some quality measures are introduced. Various quality measures are possible and their application is also dependent on the used AO system. All quantities depend on the observing wavelength λ.

Strehl ratio

The Strehl ratio relates the PSF of an AO run (and its corresponding residual wave-front) to the perfect telescope PSF, i.e., a diffraction limited PSF as presented in Section 2.1.3. While the PSF itself is the most global way to represent the quality of the observed image, the Strehl ratio gives just one number for evaluating the image quality.

The Strehl ratio (SR) is defined through

SR “ PSF_φp0q PSF_telp0q,

where PSFφ is the point spread function related to the (residual) atmospheric aber-rations resulting in a wave phase φ and PSF_tel is the diffraction limited PSF of the telescope, cf., e.g., [139, 110]. One should note that the evaluation at the origin of PSFφ and PSFtel gives a relation between their respective peaks, if they are per-fectly centered. However, due to aberrations, the peak of PSF_φ might be slightly of the center leading to a much lower Strehl ratio. Note furthermore that 0 ď SR ď1, where the equality on the right is obtained only with a perfect atmospheric correction.

The above formula is rather complicated to evaluate as one needs the full PSF of an AO run. For good corrections, related to high Strehl ratios, an approximation can be made, known asMar´echal criterion which holds forSRą0.1 (cf, e.g., [138, 139]). For small residual wave front phase σ² “ _|Ω¹

D|}φ¯´φ}²_L2pΩDq, where ¯φ“ _|Ω¹

D|

ş

ΩDφprqdr, i.e.

CHAPTER 2. ASTRONOMICAL ADAPTIVE OPTICS 31

the average phase over the aperture Ω_D , meaning thatσ² ă p2π{10q²rad², the Strehl ratio is well approximated by

SR«expp´σ²q.

Note that for computing the Strehl ratio, a wavefront has to be converted into phase, given in radians.

The above formulae are used for computing the so-called short exposure Strehl ratio (SE Strehl), meaning that the evaluation is done in each time frame. One can also consider averages over longer periods, e.g., several time steps (e.g., 30), or even a whole observation run, then the corresponding measure is called long exposure Strehl (LE Strehl), but still relies on evaluating the formulae above [99].

Full width at half maximum (FWHM)

An other measure for the image quality which can be obtained from the PSF is full width at half maximum (FWHM). It is given by the width of the PSF at the point where the intensity is half of the maximum intenstiy, as illustrated in Figure 2.14.

Figure 2.14: A diffraction limited PSF and its FWHM, source [149].

Encircled Energy (EE)

As the PSF describes as how the incoming energy is spread over the image, integrating it over a circle with its center in the middle describes another reasonable measure, the so-called encircled energy (EE). One can either fix the diameter of the circle or the

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fraction of energy, e.g., 50 %, that should be inside the circle. It is possible to also use ensquared energy, which just relates the fraction of energy to a box instead of a circle.

According to [163], encircled energy is important for observing faint objects where the concentration of photons is essential. In our simulations, we use it as a measure for the reconstruction quality of a GLAO system in Section 5.3.3.