Question #43549

A thin cylindrical glass disk of radius $R$ and initial uniform thickness $t_0$ ($t_0 \ll R$) is placed in a vacuum. The disk has thermal conductivity $k$, refractive index $n_0$ at ambient temperature $T_0$, coefficient of linear thermal expansion $\alpha$, and thermo-optic coefficient $\beta = \frac{dn}{dT}$. A uniform laser beam of total power $P$ and radius $R$ is incident normally on the disk. The glass absorbs the power uniformly throughout its volume. The cylindrical edge of the disk ($r=R$) is maintained at $T_0$, and heat loss from the flat faces is negligible. Assuming steady-state conditions, the disk acts as a thermal lens due to the radial temperature gradient. Find its effective focal length $f$.