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) 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 Path difference at point P at distance x from O is x_P = (S₀ S₁ + S₁ P) - (S₀ S₂ + S₂ P) S₁ P = D^2 + (x-d)^2 D + (x-d)^2 2D S₂ P = D^2 + x^2 D + x^2 2D x_P = (D + d^2 2D + D + (x-d)^2 2D ) - (D + D + x^2 2D ) x_P = d^2 + x^2 - 2xd + d^2 - x^2 2D = d(d - x) D For a bright fringe, x_P = n d(d - x)