A research telescope requires its primary mirror to behave as an ideal specular reflector for visible light centered at wavelength λ0=550nm. The manufacturer can polish the mirror surface so that the difference in height between adjacent peaks and valleys does not exceed hmax.
(a) Using wave-based reasoning, derive the maximum allowable phase difference between light reflected from a peak and from an adjacent valley if the mirror is to preserve constructive interference at the telescope focus.
(b) Convert your phase requirement from part (a) into an inequality that relates hmax to the wavelength λ0.
(c) Calculate the numerical value of hmax that satisfies the criterion for λ0=550nm when the allowable phase difference is chosen to be one-eighth of a wavelength.
(d) Explain why an analysis based solely on ray optics would be insufficient to set this surface-quality specification, even though a ray diagram might correctly predict the overall image formation of the telescope.