If a finite projective plane $\pi_1$ of order $m$ contains, as a sub plane, a finite projective plane $\pi_2$ of order $n$, then $m \geq n^2$ with equality holding only in the case of a Baer sub plane. Otherwise $m \geq n^2 + n$. (This is a theorem of Bruck that can be found in Hall's Group Theory book, I believe.) My question is, how nearly can this bound be achieved?
For example, we have cases for $m=9$ and $n=2$, where $\pi_1$ is one of the non-Desaurgian planes of order $9$, and $\pi_2$ is the Fano plane. Is anything known about how close we can come to the bound in the general case? Has the bound been improved?