Diffraction Limited Mode

In diffraction limited mode with adaptive optics the PSF is approximately composed of two functions:
  1. The core: This is an airy function of the telescope

    IAiry(r) $\displaystyle \sim$ $\displaystyle {\frac{{D^2}}{{\uplambda^2}}}$$\displaystyle \left(\vphantom{\frac{2\cdot \mathrm{Bessel}(x)}{x} }\right.$$\displaystyle {\frac{{2\cdot \mathrm{Bessel}(x)}}{{x}}}$$\displaystyle \left.\vphantom{\frac{2\cdot \mathrm{Bessel}(x)}{x} }\right)^{2}_{}$ (6)

    D : diameter of the telescope mirror
    $ \upmu$ : observing wavelength
    Bessel : the the first kind Bessel function of the order 1.
  2. The halo: It is given by a Moffat function

    IMoffat(r) = I$\scriptstyle \alpha$,$\scriptstyle \beta$(r) = $\displaystyle {\frac{{\beta -1}}{{\pi \alpha^2}}}$$\displaystyle \left(\vphantom{1+\frac{r^2}{\alpha^2} }\right.$1 + $\displaystyle {\frac{{r^2}}{{\alpha^2}}}$$\displaystyle \left.\vphantom{1+\frac{r^2}{\alpha^2} }\right)^{{-\beta}}_{}$ (7)


    I$\scriptstyle \alpha$,$\scriptstyle \beta$(r) : Intensity at the distance r with parameters $ \alpha$ and $ \beta$
    $ \alpha$ : Parameter is used to fix the FWHM for a given $ \beta$
    $ \beta$ : Parameter to fix the amount of light in the lobes
    r : Distance to the center (r = $ \sqrt{{x^2+y^2}}$)
    FWHM : 2$ \alpha$$ \sqrt{{2^{1/\beta}-1}}$

An example of such combined PSF is shown in Figure 1.

Figure 1: Simplified description of an observed AO-PSF: It is built by a core (Airy function) and a halo (Moffat function).
\includegraphics[width=0.77\textwidth]{pictures/psf_ao}
For comparing the peak intensity of an ideal diffraction-limited optical system with a real system the Strehl parameter was introduced.

Strehl = $\displaystyle {\frac{{I\mathrm{_{obs}}}}{{I\mathrm{_{theo}}}}}$ (8)

This parameter is the ratio of the observed peak intensity at the detection plane of a telescope or other imaging system from a point source compared to the theoretical maximum peak intensity of a perfect imaging system working at the diffraction limit. For calculating the fraction of the halo and core component to achieve a certain strehl ratio the parameter F0 is introduced in this ETC. It has to fulfill the following equation:


Strehl = $\displaystyle {\frac{{I\mathrm{_{obs}}}}{{I\mathrm{_{theo}}}}}$ = $\displaystyle {\frac{{F0\cdot I_{\mathrm{Airy}}(0) + (1-F0)\cdot I_{\mathrm{Moffat}}(0)}}{{I_{\mathrm{Airy}}(0)}}}$ (9)
$\displaystyle \rightarrow$ F0 = $\displaystyle {\frac{{I_{\mathrm{Moffat}}(0) - \mathrm{Strehl}\cdot I_{\mathrm{Airy}}(0)}}{{I_{\mathrm{Moffat}}(0)-I_{\mathrm{Airy}}(0)}}}$ (10)

In this mode the SNR is calculated for a disk with a radius of twice the radius of the airy disk.

Andre Germeroth 2014-07-17