Short exposure optical imaging through the atmosphere

It is first useful to give a brief overview of the basic theory of optical propagation through the atmosphere. In the standard classical theory, light is treated as an oscillation in a field $\psi$. For monochromatic plane waves arriving from a distant point source with wave-vector $\mathbf{k}$:

$$\psi_{0} \left(\mathbf{r},t\right) = Ae^{i\left (\phi_{o} + 2\pi\nu t + \mathbf{k}\cdot\mathbf{r} \right )}$$ (1.1)

where $\psi_{0}$ is the complex field at position $\mathbf{r}$ and time $t$, with real and imaginary parts corresponding to the electric and magnetic field components, $\phi_{o}$ represents a phase offset, $\nu$ is the frequency of the light determined by $\nu=c\left \vert \mathbf{k} \right \vert / \left ( 2 \pi \right )$, and $A$ is the amplitude of the light.

The photon flux in this case is proportional to the square of the amplitude $A$, and the optical phase corresponds to the complex argument of $\psi_{0}$. As wavefronts pass through the Earth's atmosphere they may be perturbed by refractive index variations in the atmosphere. Figure 1.2 shows schematically a turbulent layer in the Earth's atmosphere perturbing planar wavefronts before they enter a telescope. The perturbed wavefront $\psi_{p}$ may be related at any given instant to the original planar wavefront $\psi_{0} \left(\mathbf{r}\right)$ in the following way:

$$\psi_{p} \left(\mathbf{r}\right) = \left ( \chi_{a} \left(\m... ...ft(\mathbf{r}\right)}\right ) \psi_{0} \left(\mathbf{r}\right)$$ (1.2)

where $\chi_{a} \left(\mathbf{r}\right)$ represents the fractional change in wavefront amplitude and $\phi_{a} \left(\mathbf{r}\right)$ is the change in wavefront phase introduced by the atmosphere. It is important to emphasise that $\chi_{a} \left(\mathbf{r}\right)$ and $\phi_{a} \left(\mathbf{r}\right)$ describe the effect of the Earth's atmosphere, and the timescales for any changes in these functions will be set by the speed of refractive index fluctuations in the atmosphere.

Figure 1.2: Schematic diagram illustrating how optical wavefronts from a distant star may be perturbed by a turbulent layer in the atmosphere. The vertical scale of the wavefronts plotted is highly exaggerated.