Root fence to keep poplar tree roots from new vegetable garden?
 

I just cut out a large ares of sod for a new vegetable garden. Digging a hole in
the new garden area, i find lots of tree roots, in the 1/2"-1" diameter range.
There are two 80' poplar trees about 50' south of this new bed, plus closer
apple (30' away) and plum trees (15' away). I suspect the poplar roots are the
biggest problem.

I'm wondering if a root fence could be effective in stopping this invasion. If I
dig a ditch around the edge of the bed, cutting the roots I find, and insert
plastic root fence material around the bed, will that have any significant
effect in preventing the roots from re-invading after I cut the ones I hit? How
deep would a root fence need to be to really work? Or, will the roots just dive
under it and come back to the surface?

Any good reference suggestions?