For an optical arrangement as depicted in the figure, the distance D is significantly greater than the slit separation d . Assume d is set to the minimum value required to produce a dark fringe at point O . Determine the fringe width of the interference pattern.

Step-by-step solution

Path difference at point O is x_O = (S₀ S₁ + S₁ O) - (S₀ S₂ + S₂ O) For D d , the distances are: S₀ S₁ = S₁ O = D^2 + d^2 D + d^2 2D S₀ S₂ = S₂ O = D x_O = 2 (D + d^2 2D ) - 2D = d^2 D For a dark fringe at O with minimum d , x_O = 2 d^2 D = 2 = 2d^2 D The fringe width on the screen for sources separated by d at distance D is = D d = ( 2d^2 D ) D d = 2d