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

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

   $$\alpha \beta {\frac{{\beta -1}}{{\pi \alpha^2}}} \left(\vphantom{1+\frac{r^2}{\alpha^2} }\right. {\frac{{r^2}}{{\alpha^2}}} \left.\vphantom{1+\frac{r^2}{\alpha^2} }\right)^{{-\beta}}_{}$$ (7)

   
   
   
   

   I$\alpha$,$\beta$(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).

For comparing the peak intensity of an ideal diffraction-limited optical system with a real system the Strehl parameter was introduced.

$${\frac{{I\mathrm{_{obs}}}}{{I\mathrm{_{theo}}}}}$$ (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:

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

$$\rightarrow {\frac{{I_{\mathrm{Moffat}}(0) - \mathrm{Strehl}\cdot I_{\mathrm{Airy}}(0)}}{{I_{\mathrm{Moffat}}(0)-I_{\mathrm{Airy}}(0)}}}$$ = (10)

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