= Solution
The <paraxial approximation> restricts <light rays> to small slopes relative to the <optical axis>, so $\sin\beta\simeq\tan\beta\simeq\beta$ and geometric lengths can be expanded to second order in height. Treat the source's <optical lens> as a <thin lens> located at $x=0$, with $|y|\ll u,|v|$ and small surface slopes.
For positive real conjugate distances $u,v$, the collapsed-plane geometric path has length
$$
S(y)=\sqrt{u^2+y^2}+\sqrt{v^2+y^2}
\simeq u+v+\frac{y^2}{2}\left(\frac1u+\frac1v\right).
$$
To thin-element accuracy, a segment $s(y)=s_0-\alpha y^2$ of this path lies inside the <optical lens>, replacing the same length of air. The geometric air path is $S-s$, the material path is $s$, and the <optical path length> and travel time are therefore
$$
\mathcal L(y)=S(y)+(n-1)s(y),\qquad
ct(y)\simeq u+v+(n-1)s_0
+y^2\left[\frac1{2u}+\frac1{2v}-(n-1)\alpha\right].
$$
This distinguishes geometric length from <optical path length>; adding $ns$ to $S$ without subtracting the replaced air would count that segment twice.
For a nonaxial <light ray>, <Fermat's principle> gives $d\mathcal L/dy=0$ and hence
$$
\frac1u+\frac1v=2(n-1)\alpha.
$$
The <focal length> is obtained with the incident object at infinity:
$$
\boxed{\frac1f=2(n-1)\alpha,\qquad f=\frac1{2(n-1)\alpha}.}
$$
The constant thickness $s_0$ supplies a common delay but does not change the leading <focal length>. Since $\alpha$ has units of inverse <length>, this formula has the correct dimensions. These are <thin lens> results; <paraxial approximation> alone does not remove finite-thickness corrections in a general <thick lens>.
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