A researcher wishes to measure the separation of two slits in a double-slit slide that will be used for precision optical experiments. White light is blocked so that only a monochromatic beam of known wavelength λ=520nm illuminates the slits. The interference pattern is projected onto a screen that is L=0.60m from the slits. Because the screen is relatively close, the researcher decides to avoid small-angle approximations.
The first-order bright fringes appear at vertical positions y+1=4.20mm above and y−1=−4.12mm below the central maximum.
(a) Derive an expression for the slit separation d in terms of the measured angle θ1 corresponding to the first-order maximum.
(b) Using the exact trigonometric relationship between sinθ1 and the measured geometry, calculate d. Present your answer in micrometres to three significant figures.
(c) Re-calculate d with the small-angle approximation sinθ≈tanθ≈y/L. Determine the absolute difference between the two values of d and compute the percent error that would arise from using the approximation.
(d) Explain why the approximation becomes less accurate as the screen is moved closer to the slits, citing quantitative evidence from your results.