Question #43128

A thin equiconvex lens is made of a material with refractive index $n_0 = 1.5$ at a uniform temperature $T_{\text{ref}}$. At $T_{\text{ref}}$, the lens has a central thickness $t_0 = 0.5 \text{ cm}$, an aperture radius $a = 1.0 \text{ cm}$, and a paraxial focal length $f_0 = 20.0 \text{ cm}$ in vacuum. The lens is subjected to a steady radial temperature profile $\Delta T(r) = T_0 \left(\frac{r}{a}\right)^2$, where $r$ is the radial distance from the optical axis and $T_0 = 100 \text{ K}$. The material has a linear thermal expansion coefficient $\alpha = 8.0 \times 10^{-5} \text{ K}^{-1}$ and a thermo-optic coefficient $\frac{dn}{dT} = \beta = 6.0 \times 10^{-5} \text{ K}^{-1}$. Assume the lens expands locally such that its thickness at distance $r$ becomes $t'(r) = t(r)[1 + \alpha \Delta T(r)]$, and ignore any radial displacement of the material. Calculate the new effective paraxial focal length $f$ of the lens.