Image-position error associated with a quadrant detector

An exact expression is obtained that evaluates the error associated with angular-position measurement of an incoherently illuminated object when a quadrant detector is used in the image plane of an optical system. The accuracy of such a measurement is inversely proportional to the signal-to-noise voltage ratio associated with the four quadrants summed to act as a single detector. In the particular case of a circular object, the rms angular measurement error is given by the expression σθ = π[(3/16)2 + (n/8)2]1/2(λ/D)/SNRv, where n is the angular subtense of the object divided by the diffraction angle λ/D, SNRv is the signal-to-noise voltage ratio, λ is the wavelength of the light used, and D is the diameter of the limiting aperture in the optical system under consideration.