The performance of the adaptive optics (AO) is evaluated by simulating point spread functions (PSFs) using ESO's numerical AO simulation tool OCTOPUS. In future, we also plan to use L. Jolissaint's analytical PSF simulation tool PAOLA.

For each simulated PSF we also provide the paramaters of an analytic fit to its radial intensity profile. This was obtained using eltpsffit, a custom-built tool to fit the numerically simulated PSFs with a combination of various analytical functions. The purpose of this tool is to obtain speckle-free, more manageable representations of these (very large) PSF images.

Below we provide a brief description of the simulations. In addition, here is some general introductory and supplementary information on AO (more), types of AO, AO simulations and atmospheric turbulence which may be helpful.

Finally, please note that the MAORY consortium has also made their simulated MCAO PSFs available online.

ESO PSF simulations

Here we give a basic description of ESO's numerical AO simulation tool OCTOPUS. Additional details are provided by Le Louarn et al. (2005) and Le Louarn et al. (2004).

The atmosphere is represented by a small number (≈ 10) of infinitely thin turbulent layers which act as phase screens. Each screen is set up randomly and independently from one another according to a von Karman power spectrum of the refractive index fluctuations, i.e. a power-law spectrum with index -11/3 curtailed at finite inner and outer scales. The normalisation of each layer's power spectrum is given by the atmosphere's Cn2 profile, which describes the distribution of the turbulence in the atmosphere as a function oh height above the telescope.

The light from the (natural or laser) guide stars (NGS or LGS) is propagated to the wavefront sensors (WFS) using the geometrical optics approximation, i.e. tracking only phase fluctuations but not amplitude fluctuations (scintillation) of the light. The finite height of any LGS is taken into account by propagating their light in a conic fashion, rather than cylindrically as for real astronomical sources (like NGS). All WFS are assumed to be of the Shack-Hartmann type, i.e. an array of lenslets is used to produce multiple images of the guide star, each in its own sub-aperture on the WFS detector. The images on the WFS are formed by calculating the squared modulus of the Fourier transform of the incoming complex amplitude at the location of each lenslet. These images are then degraded by adding photon noise according to the guide star brightness and WFS integration time, as well as read-out noise. Subsequently, the centroids of the guide star images in all of the sub-apertures are measured, and all measurements from all WFS are assembled into a single vector. (Note that the calculation of the centroids ignores LGS spot elongation because new algorithms are currently being studied which will deal with this effect optimally.)

The position measurement vector is then multplied with the AO system's pre-computed command matrix. The command matrix is the inverse of the interaction matrix which describes how the guide star positions in each of the WFS sub-apertures respond to a unit 'push' (or 'pull') from each of the deformable mirror's (DM) actuators. The result of multiplying the positional measurement vector with the command matrix is therefore a vector describing the positions of the DM's actuators that are required to compensate for the measured wavefront distortions. The DM is modeled as a special phase screen. Its shape (given by the actuators' positions and the assumed influence function) is subtracted from the phase of a wavefront coming in from the desired direction in the field of view (and propageted cylindrically). The resulting wavefront is then Fourier transformed to create a quasi-instantaneous or short exposure PSF, which is recorded.

To account for the temporal evolution of the atmosphere the above process is then repeated many times. Each iteration starts by shifting the phase screens representing the atmosphere by an amount determined by each layer's wind speed and the duration of one iteration in real time. Then the whole process of guide star wavefront propagation, wavefront sensing, DM shape calculation and short exposure PSF generation is repeated. Actually, the DM shape that is subtracted from the wavefront in any given iteration is not the shape computed in the same iteration, but rather the shape computed several iterations earlier. This delay accounts for the time required to integrate on the guide star (= 1 iteration), read out the WFS detectors, compute the actuator positions, and actuate the DM.

Each iteration results in a short exposure PSF. All of these are summed together to generate the final, 'long' exposure PSF. However, since the frame rate of an AO system is typically 500 – 1000 Hz it takes thousands of iterations to cover just a few seconds of real time. Herein lies the main limitation of these simulations: since they are computationally so expensive it is impossible to generate truly long exposure PSFs (covering many minutes of real time) and one is limited to a few seconds of exposure time. The result is that residual speckles are still visible in the final PSFs which, of course, would not be present in real astronomical observations covering many minutes. This is obviously an undesirable feature when using these PSFs to simulate astronomical images. One way to overcome this limitation is to fit the simulated PSFs with a suitably chosen analytical function, or a combination of several different functions, in order to obtain a speckle-free representation.

Note that the above simulation process describes a Monte Carlo simulation: two realisations of the same PSF will differ from one another because of the random nature of the simulated atmosphere and because of the noise associated with the guide star images.

ESO simulation parameters

Obviously, there a large number of input parameters that are required for the simulations. Due to the computational cost of these simulations it is impossible to explore this parameter space comprehensively. Hence, only a relatively small number of input parameters (such as, e.g., the wavelength of observation) were varied to create different PSFs (see below). All other parameters were held constant for all simulations. These parameters and their values are listed below.


    Fried parameter r0
    0.13 m at 0.5 μm
    0.8" at 0.5 μm
    Turbulence power spectrum
    von Karman
    Outer scale L0
    25 m
    Inner scale l0
    pixel size
    Number of turbulent layers

    Parameters of layers
         Height [m]      Fractional Cn2      Windspeed [m/s]
    0 0.335 12.1
    600 0.223 8.6
    1200 0.112 18.6
    2500 0.090 12.4
    5000 0.080 8.0
    9000 0.052 33.7
    11500 0.045 23.2
    12800 0.034 22.2
    14500 0.019 8.0
    18500 0.011 10.0

    Laser guide stars



    Number of LGS
    LGS brightness
    LGS positions [arcmin from field centre, i=0…4]
    x = 3 cos( i×72°)
    y = 3 sin( i×72°)
    0.75 cos( i×72°), 0
    0.75 sin( i×72°), 0

    Primary mirror

    Size and geometry
    See the telescope section.

    Wavefront sensors

    Number of WFS
    equal to number of LGS
    Number of sub-apertures per WFS
    84 × 84
    Number of CCD pixel per sub-aperture
    none (infinite flux, no read-out noise)
    spot elongation

    Deformable mirrors

    Number of DMs
    Number of actuators per DM
    85 × 85
    Actuator stroke
    Conjugation height
    0 m
    Tilt with respect to layers
    Influence function
    linear spline

    Other parameters

    Frame rate
    500 Hz
    Number of iterations
    Total integration time
    4 s
    Delay time
    3 iterations


ESO PSF fitting

In order to obtain speckle-free, more manageable representations of the PSF images, the radial profiles of all PSFs in the database have been fit with combinations of analytical functions using the custom-built tool eltpsffit. We only performed 1D fits of the PSF profiles, not 2D fits of the whole images, because generally full 2D fitting did not result in a significantly improved representation of the PSF.

High-Strehl PSFs were typically fit with four to five components: an (obstructed) Airy disk to represent the diffraction limited core and additional Lorentzian or Moffat function components to represent the halo. Low-Strehl PSFs usually required only two to three Lorentzian or Moffat components. Our (somewhat arbitrary) aim was to obtain fits that lie within 5% of the data at all radii (not just in the centre). This was generally achieved except for the mid-IR LTAO PSFs which are much more difficult to fit.

Note that these fits are not perfect representations of the PSFs. Different choices for the fitting weights, number or type of the fit components, etc., may well lead to more accurate descriptions, especially when considering only a restricted range of radii. Hence, depending on your application, you may wish to use eltpsffit to produce your own fits.

ESO PSF database

All PSFs are stored as fits images. The PSF image database is organised in a directory tree structure, where the levels correspond to those input parameters that were varied to create different PSFs:

  • type of AO
  • zenith distance
  • wavelength
  • size of the field of view
  • position within field of view

At a given level of the tree you will find one directory for each value of the level's corresponding parameter for which simulations were performed. The actual PSF images are contained in the lowest level directories. To find the PSF that corresponds to a particular combination of parameter values simply work your way down the directory tree.

Note that due to the computational cost of these simulations it is not possible to homogeneously populate even this relatively small 5-dimensional parameter space (let alone the full parameter space, see above). PSFs were generated only for those combinations of parameter values that have so far been deemed interesting or necessary. Hence some parts of parameter space are populated more densely than others, while other parts are entirely empty. Those combinations of parameter values that have so far not been simulated do not have corresponding branches in the database directory tree.

An existing branch usually only has one PSF image in its lowest level directory. However, for a few combinations of parameter values the simulations have been repeated, resulting in more than one realisation of the same PSF. In these cases the lowest level directory contains more than one PSF image. As explained above, all PSFs have been fitted with analytical functions in order to obtain a speckle-free representation. The files containing the results of the fits are also stored in the lowest level directories. These files have the same root filename as the corresponding PSF image but different extensions. Their contents are explained here.

Note that the fits header of each PSF image contains a record of the above parameter values that were used to simulate this PSF. The header also contains the usual basic image information such as the image's pixel size in mas (keywords CDELT1 and CDELT2).

The AO type 'NOAO' refers to the seeing-limited case (read: no AO). Zenith distance is given in degree. The wavelength of observation is given in m in the image headers, but is listed by the corresponding filter name in the database directory tree. The parameter 'size of the field of view' only applies to GLAO PSFs and is given in arcmin. This level is absent in the LTAO and NOAO parts of the directory tree. The parameter 'position whithin field of view' consists of an x and a y coordinate given in arcsec for LTAO and NOAO and in units of the size of the field of view for GLAO.

Below is the database directory tree. Click on a folder icon to expand or collapse that folder. Click on the folder name itself to go to that folder.

Expand all | Collapse all

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ESO PSF properties

These plots show how the PSF depends on the various parameters above.

PAOLA PSF simulations

PAOLA is an analytical PSF modeling tool developed by L. Jolissaint. The reasons to explore anlaytical PSF simulations in addition to ESO's numerical simulations include: (i) they are free of speckle noise; (ii) the code is much faster and so one can explore a larger parameter space; (iii) one wants to understand the sensitivity of any DRM conclusions to the details of the PSF.

The fundamentals of analytical PSF modelling are described by Jolissaint et al. (2006).

So far, this work has not been completed...