Example short exposure images

I will first investigate imaging performance in more detail using simulations of Kolmogorov atmospheres. For these simulations I have chosen to ignore amplitude fluctuations contained in the atmospheric term $\chi_{a} \left(\mathbf{r}\right)$ entirely - this corresponds to the case where atmospheric refractive index perturbations are only found very close to the aperture plane of the telescope. This is achieved in the simulations simply by setting:

$$\forall \mathbf{r}:\mbox{ }\chi_{a} \left(\mathbf{r}\right)=1$$ (1.6)

The effect of the finite aperture size of the telescope can be simulated by setting the wavefront amplitude to zero everywhere in the aperture plane except where the light path to the primary mirror is unobstructed. This can be achieved most easily by defining a function $\chi_{t} \left(\mathbf{r}\right)$ which describes the effect of the telescope aperture plane coverage on the wavefronts in the same way that the effect of the atmosphere is described by $\chi_{a} \left(\mathbf{r}\right)$. The value of $\chi_{t} \left(\mathbf{r}\right)$ will be zero beyond the edge of the primary mirror and anywhere the primary is obstructed, but unity elsewhere. For the simple case of a circular primary mirror of radius $r_{p}$ without secondary obstruction:

$$\chi_{t}\left ( r \right ) = \left\{ \begin{array}{ll} 1 &... ...ft \vert \mathbf{r} \right \vert > r_{p}$} \end{array}\right.$$ (1.7)

Phase perturbations introduced into the wavefronts by aberrations in the telescope can be described by a function $\phi_{t} \left(\mathbf{r}\right)$ in similar way, resulting in wave-function $\psi_{p}'$ given by:

$$\psi_{p}' \left(\mathbf{r}\right) = \left ( \chi_{a} \left(\... ...r}\right) \right ] } \right ) \psi_{0} \left(\mathbf{r}\right)$$ (1.8)

I will begin with the simple case of a telescope which is free of optical aberrations observing in a narrow wavelength band. The perturbed wave-function reaching the telescope aperture for this case is given by setting $\phi_{t} \left(\mathbf{r}\right)\equiv 0$ in Equation 1.8. Combined with Equation 1.6 this gives:

$$\psi_{p} \left(\mathbf{r}\right) = \left ( \chi_{t} \left(\m... ...t(\mathbf{r}\right)} \right ) \psi_{0} \left(\mathbf{r}\right)$$ (1.9)

For the simulations a long-period pseudo-random number generator was used to produce two-dimensional arrays containing discrete values of $\phi_{a} \left(\mathbf{r}\right)$, having the second order structure function defined by Equation 1.4, using a standard algorithm provided by Keen (1999). The time evolution of $\phi_{a} \left(\mathbf{r}\right)$ was ignored, as I was interested in the instantaneous imaging performance (the case for short exposures). Arrays of $\psi_{p} \left(\mathbf{r}\right)$ were then generated corresponding to the wave-function provided by a distant point source after passing through the atmospheric phase perturbations and the telescope aperture using Equation 1.9. These arrays were Fourier transformed using a standard Fast Fourier Transform (FFT) routine to provide images of the point source as seen through the atmosphere and telescope. The image of a point source through an optical system is called the point-spread-function (PSF) of the optical system. For our simple optical arrangement with phase perturbations very close to the aperture plane, the response of the system to extended sources of incoherent light is simply the convolution of the PSF with a perfect image of the extended source.

Figure 1.3 shows simulated PSFs for three atmospheric realisations having the same $r_{0}$ and image scales but with different telescope diameters. There are two distinct regimes for the cases of large (diameter $d\gg r_{0}$) and small ($d\sim r_{0}$) telescopes. Figure 1.3a is a typical PSF from a telescope of diameter $d=20r_{0}$. The image is broken into a large number of speckles, which are randomly distributed over a circular region of the image with angular diameter $\sim \frac{\lambda}{r_{0}}$, where $\lambda$ represents the wavelength. With the slightly smaller aperture shown in Figure 1.3b the individual speckles are larger - this is because the typical angular diameter for such speckles is $\sim1.22\frac{\lambda}{d}$, equal to the diameter of the PSF in the absence of atmospheric phase perturbations for a telescope of the same diameter $d$ (i.e. a diffraction-limited PSF). For the small aperture size shown in Figure 1.3c the shape of the instantaneous PSF deviates little from the diffraction-limited PSF given by a telescope of this diameter. The first Airy ring is partially visible around the central peak.

Figure 1.3: Typical short exposures through: a) a $20r_{0}$ aperture; b) a $7r_{0}$ aperture; and c) a $2r_{0}$ aperture. All three are plotted with the same image scale but have different greyscales.

Real astronomical images of small fields obtained through the atmosphere will correspond to an image of the sky brightness distribution convolved with the PSF for the telescope and atmosphere. The perturbations introduced by the atmosphere change on timescales of a few milliseconds (known as the atmospheric coherence time). If the exposure time for imaging is shorter than the atmospheric coherence time, and the telescope is free of optical aberrations, then Figures 1.3a--c will be representative of the typical PSFs observed. The random distribution of speckles found in the short exposure PSFs of Figures 1.3a and 1.3b will have the effect of introducing random noise at high spatial frequencies into the images, making individual short exposures such as these of little direct use for high resolution astronomy. Figure 1.3c is dominated by a relatively uniform bright core, and as such will provide images with relatively high signal-to-noise. Unfortunately the broad nature of the PSF core severely limits the image resolution which can be obtained with such a small aperture.