it's perhaps worth noting that Larry Niven's ringworld blocks about 0.5% of its star's output. It's an interesting read because at a certain age the thing feels doable, but also Niven is at some pains to impress upon the reader how impossibly large it is.
1/
1PW of light can accelerate a mass up to 670t with a 1g starting acceleration and get it relativistic in under a year.
100,000,000PW reflected could reasonably send 6.7 gigaton of total mass (1-5Gt payload) out of the solar system a year at relativistic speeds.
The Sun puts out over 380,000,000,000PW.
3% of its output at 30% pumping efficiency, 90% beam targeting efficiency, and 25% reflectivity (75% transparent ultralightweight sail) would be good for over 50Gt/yr of total mass, at least 10Gt/yr of which could be payload.
Iow sending billions of biological human beings out of the solar system a year at >0.5c is quite within the reach of laser sails.
I don't know of a more promising technology for this.