Zernike annular polynomials for imaging systems with annular pupils

The aberrations of imaging systems with uniformly illuminated annular pupils are discussed in terms of a complete set of polynomials that are orthogonal over an annular region. These polynomials, which we call Zernike annular polynomials, are similar to the Zernike circle polynomials and reduce to them as the annulus approaches the full circle. The Zernike-polynomial expansion of an aberration function is compared with its power-series expansion. The orthogonal aberrations given by Zernike annular polynomials describe how a higher-order classical aberration of a power-series expansion is balanced with one or more lower-order classical aberrations to minimize its variance. It is shown that as the obscuration ratio increases, the standard deviation of an orthogonal as well as a classical primary aberration decreases in the case of spherical aberration and field curvature and increases in the case of coma, astigmatism, and distortion. The only exception is the case of orthogonal coma, for which the standard deviation first increases and then decreases. The orthogonal aberrations for nonuniformly illuminated annular pupils are also considered, and, as an example, Gaussian illumination is discussed. It is shown that the standard deviation of an orthogonal primary aberration for a given amount of the corresponding classical aberration is somewhat smaller for a Gaussian pupil than that for a uniformly illuminated pupil.