A manufacturer extrudes a transparent polymer sheet of index of refraction n=1.60 so that its thickness increases linearly from one edge to the other. The sheet is illuminated from above by monochromatic light of wavelength λ0=500 nm in air at normal incidence. At the thinner edge a bright reflected fringe corresponding to order m=0 is observed, and at the thicker edge the bright fringe corresponding to order m=30 is observed.
(a) Derive an expression relating the sheet thickness t to the interference order m for the reflected bright fringes under the given conditions.
(b) Using the information provided, calculate the thickness difference between the two edges of the sheet. Express your answer in micrometers.
(c) If the illumination wavelength is changed to λ0=450 nm while all other conditions remain the same, determine the interference order number that will now appear bright at the thicker edge. Show your reasoning.