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Military grade lasers shooting down incoming missiles have been in the news since Ronald Reagan was the US president. The problems, as I recall are mostly about diffusion in atmosphere.
The Kessler syndrome suggests at some point the amount of debris in LEO is going to be so great it creates significant issues.
When LEO is so full of debris that you can no longer find a safe window to launch through, could you use laser mounted on your rocket to clear a launch path?
Well, for a laser to help in this situation, you'd need to vaporize the debris. That means heating the entire piece to its evaporation temperature. This needs either a very powerful laser, or enough dwell time at a lower energy level.
Both are difficult to achieve. For high laser power, you're talking about the sort of power that is generated by the first stage engines, plus tons of laser equipment.
At lower power, you need to account for the fact that the debris you're worried about has a speed on the order of kilometers per second relative to the launch vehicle. This makes it tricky to target the debris accurately. If you need 5 seconds dwell time for an object 10 cm across, you need to hit it when it's 5-50 km away and keep hitting it continuously. Objects that small are very difficult to hit.
You'd be better off putting the laser on a satellite, where it can shoot at targets at leisure, and not care how long it takes. That way you can also do selective ablation on larger pieces, and produce retrograde thrust to deorbit them. Power's still a problem though.
