Typo! Also, probably best to call this Resolution like the other similar inputs

Simpler IMO to just use expand * expand here

Instead of dividing by verts - 1 in multiple places, you can define const float delta = 1.0f / (verts - 1); and then multiply by it where you divided before.
The other benefit is that multiplying is much faster than dividing, sometimes noticeably so.

You can pass delta * i to fall_off instead of the index as well.

Here's how I'd write this given the above comment: const float z = std::pow(1.0f - delta, exponent) * depth;
Usually it's more clear to declare more than one const variable than to change the same one over multiple lines.

I think this (radius / max_radius) value is constant for the entire loop, so you could pull that out.