The mixture a pure liquid and a solution in a long vertical column (i.e, horizontal dimensions << vertical dimensions) produces diffusion of solute particles and
hence a refractive index gradient along the vertical dimension. A ray of light entering the column at right angles to the vertical deviates from its original path.
Find the deviation in travelling a horizontal distance d << h, the height of the column.
Let the height of the long vertical column with transparent liquid be h and dx be the thickness
The angle at which the ray AB enters is θ
Let y be the new height of the liquid
(θ + d θ) is the emerging angle while (y + dy) is the height.
From Snell’s law,
μ(y) sin θ = μ (y + dy) sin (θ + d θ)
After integration we get,
θ = -1/ μ d μ/dy d
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