The Computation of Diffraction Patterns in the Presence of Aberrations

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Abstract
The diffraction integral for the disturbance produced in the image plane normal to the optical axis by an extra-axial pencil has been shown to lead to a Fourier transform provided the exit pupil surface is taken to be that of the reference sphere. It has been shown also that, except for small aperture and field sizes, the effect on the wave-front aberration of a shift of the image plane is not represented with sufficient accuracy merely by a term proportional to the aperture squared. Both of these results have been respected in formulating a numerical technique for the calculation of point spread functions. The diffraction integral is evaluated in polar coordinates, and is such that no error is made in approximating the domain of the exit pupil in cases where this may be represented by an ellipse. A study of the accuracy obtained has shown that, if each quadrant of the pupil is divided into a 20 × 20 mesh of elementary areas, the error in the intensity is not expected to exceed 0·8 per cent of the intensity at the focus of a diffraction limited system. The method takes account of the first derivatives of the wave aberration at each mesh point, and the results are therefore expected to be more accurate than those obtained by merely replacing the integral by a simple sum. Results are given for a case of primary and secondary coma, and of a study of the influence of secondary spectrum and spherical aberration on the images formed by 2 mm achromatic microscope objectives of numerical aperture equal to 1·40.

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