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Certain configurations of massive structures projected along the line ofsight maximize the number of detections of gravitationally lensed $z\sim10$galaxies. We characterize such lines of sight with the \'etendue $\sigma_\mu$,the area in the source plane magnified over some threshold $\mu$. We use theMillennium I and Millennium XXL cosmological simulations to determine thefrequency of high $\sigma_\mu$ beams on the sky, their properties, andefficient selection criteria. We define the best beams as having$\sigma_{\mu>3} >2000$ arcsec$^2$, for a $z\sim10$ source plane, and predict$477 \pm 21$ such beams on the sky. The total mass in the beam and$\sigma_{\mu>3}$ are strongly correlated. After controlling for total mass, wefind a significant residual correlation between $\sigma_{\mu>3}$ and the numberof cluster-scale halos ($>10^{14} M_\odot h^{-1}$) in the beam. Beams with$\sigma_{\mu>3} >2000$ arcsec$^2$, which should be best at lensing $z\sim10$galaxies, are ten times more likely to contain multiple cluster-scale halosthan a single cluster-scale halo. Beams containing an Abell 1689-like massivecluster halo often have additional structures along the line of sight,including at least one additional cluster-scale ($M_{200}>10^{14}M_\odoth^{-1}$) halo 28% of the time. Selecting beams with multiple, massivestructures will lead to enhanced detection of the most distant andintrinsically faint galaxies.

Characterizing the Best Cosmic Telescopes with the Millennium Simulations