Problem 5.218

A light filter is a plate of thickness $d$ whose absorption coefficient depends on wavelength $\lambda$ as $$x(\lambda)=\alpha\left(1-\lambda / \lambda_{0}\right)^{2} \mathrm{~cm}^{-1}$$ where $\alpha$ and $\lambda_{0}$ are constants. Find the passband $\Delta \lambda$ of this light filter, that is the band at whose edges the attenuation of light is $\eta$ times that at the wavelength $\lambda_{0} .$ The coefficient of reflection from the surfaces of the light filter is assumed to be the same at all wavelengths.