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Schematic showing how the expected numerical aperture was calculated from the ratio of the diameter of the spot size to the diameter of the ball. α can be calculated by tan(dspot/dball) = α/2. This formula applies because the maximum incidence angle of light in the setup that should be seen as a bright spot on the photonic ball should be the angle for which the specular reflection is along the microscope axis; the half angle of the numerical aperture (α) satisfies this condition.
