Is it possible to use a large lightweight sail deployed on the first stage shortly after separation from the second stage to receive exhaust thrust from the second stage engine, so as to slow the first stage down?
tl;dr: Comparing 6 to 16 m/s delta-v to the 2nd stage velocity at separation of about 2,000 m/s it seems there's just not much help.
alt-tl;dr: To quote @LorenPechtel's insightful comment:
Well, if you're using a photon drive. Otherwise it disperses so fast you gain almost nothing.
The question is clear. While I'm not a rocket scientist I'll venture an answer. I'll use a Falcon 9 1st and 2nd stage ballpark numbers as nominals for an envelope-back calculation.
Let's assume the stage separation has no large "kick" and that there's a cone at the top of the 1st stage that deflects the thrust impinging from the 2nd stage sideways. This cone magically expands to a diameter of 100 meters (50 meter radius) and manages to convert half of the force from the 2nd stage thrust that hits it into a decelerating force of the 1st stage until they are separated by 100 meters, at which point much of the 2nd stage exhaust expands so much that it starts missing the cone.
t = metric ton = 1,000 kg 2nd stage thrust: 105 kN 2nd stage mass : 97 t remaining propellant 0% 10% 20% ----------------------------- 1st stage mass: 26 65 105 t 1st stage acceleration: 2.0 0.8 0.5 m/s^2 2nd stage acceleration: 1.1 1.1 1.1 m/s^2 time to 100 m separation: 8.8 10.3 11.3 s 1st stage delta-v 16.2 8.3 5.6 m/s
Comparing 6 to 16 m/s delta-v to the total velocity at separation of about 2,000 m/s it seems there's just not much help.
Consider that the last ten seconds of the 1st stage flight had nine equivalent engines accelerating it, ten seconds of half the thrust of one engine won't do much to reverse that.