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.