14.4.2 Synthetic beam

The neat beam can be regarded as a sort of optimal clean beam: the optimal apodized point-spread function that can be designed within the limits of the Heisenberg principle. More precisely, the neat beam $\Theta _s$ is a centro-symmetric function lying in the object space H_o, and satisfying the following properties:

• The energy of  $\widehat{\Theta}_s$ is concentrated in  $\mathcal{H}_s$. In other words, $\widehat{\Theta}_s$ has to be small outside  $\mathcal{H}_s$in the mean-square sense: we impose the fraction $\chi^2$ of this energy in  $\mathcal{H}_s$ to be close to 1 (say  $\chi^2=0.98$).
• The effective support  $\mathcal{D}_s$ of $\Theta _s$ in H_o is as small as possible with respect to the choice of  $\mathcal{H}_s$and $\chi^2$. The idea is of course to have the best possible resolution.

This apodized point-spread function is thus computed on the grounds of a trade-off between resolution and efficiency, with the aid of the power method.

  

Figure: Experimental frequency coverage and frequency coverage to be synthesized  $\mathcal{H}_s$(left hand). The experimental frequency list $\mathcal{L}_e$ includes N_e=2862 frequency points. The frequency coverage to be synthesized  $\mathcal{H}_s$is centred in the Fourier grid  $\text{\boldmath $G$\unboldmath}\,\delta u$, where $\delta u = \Delta u/N$ with N=128 (here, the diameter of the circle is equal to  $40\,\delta u$). The neat beam $\Theta _s$ (right hand) represented here corresponds to the frequency coverage to be synthesized $\mathcal{H}_s$ for a given value of  $\chi ^2=0.97$. It is centred in the object grid $\text{\boldmath $G$\unboldmath}\,\delta x$ where  $\delta x=1/\Delta u$(here, the full width of $\Theta _s$ at half maximum is equal to  $5\,\delta x$).