I've spent a good amount of engineering hours doing research on this thought when I had it in the shower a few years back. My thought was a little different in that : " perhaps we could beam a laser with encoded information on a loop journey, that is meant to return to earth in X million years."

Such a trajectory might also depend on the position of earth and other objects and have a very narrow windows. As if watching a pond on a windy day and the wind eases off for a second or two yielding a glimpse of your own reflection and then to pick up again leaving a blurry pond.