A multilayer coating alters the amplitude and phase of a reflected wave front. The amplitude effects are multiplicative and well understood. We present a mathematical formalism that can be used to describe the phase effects of coating in a general case. On the basis of this formalism we have developed an analytical method of estimating the wave-front aberrations introduced by the multilayer coating. For the case of field-independent aberrations, we studied both uniform and graded multilayer coatings. For the case of field-dependent aberrations, we studied only the effects of a uniform multilayer coating. Our analysis is based on a coated plane mirror tilted with respect to an incident converging beam. Altogether we have found, up to the second order, the following aberrations: a field-dependent piston, a field-squared-dependent piston, defocus, field-independent tilt, field-independent astigmatism, and anamorphic magnification. To obtain numerical results we apply our analysis to the specific case of a plane mirror tilted 8.2 deg with respect to an incident converging beam with a numerical aperture of 0.1. We find that the magnitudes of the field-independent aberration coefficients for the graded coating are approximately ten times smaller than those for the uniform coating. We show that a coating can introduce anamorphic magnification.
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