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The generalized sine condition for an image-forming system with centering errors but allowing for one symmetry plane is derived according to the Fourier optics approach. The variation of the wave-front-aberration function associated with a small displacement of field coordinates is given. The symmetry properties of aberrations are discussed.͓(x o Ј , y o Ј)͔, coordinates at the entrance (exit) pupil with origin at Ẽ o (Ẽ o Ј).

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We calculate the complex degree of coherence for any interferometer in which there is amplitude division and simple interference. We consider the exact optical path length differences between both branches. We find that, when there is a localization surface, this degree of coherence is equivalent to the complex optical disturbance at the image of an aberrated isoplanatic optical system. Moreover, we show that the radius of the central lobe at the diffraction pattern of this equivalent system is the distance between rays when visibility is zero. This distance enables us to evaluate localization depth and adjustment tolerance.

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If one considers the interference of two extended sources and if one is able to observe fringes only at certain points in space one says that the fringes are localized on those points. As an example we show how the localized fringes may be found in a device used to observe Newton’s rings. Moreover we give a proof of the general method used to determine the localized fringes position. This proof is suitable for undergraduate optics courses. We also analyze the visibility of fringes.