A direct approach to the evaluation of the variance of the wave aberration

Maréchal's treatment of tolerance theory shows that in designing high quality systems one should aim at minimizing the variance E of the wave aberration. Since the value of E is essentially positive, a useful criterion for the whole field is the sum of the suitably weighted values of E for a typical set of image points. It is shown here that the variance E (for both the axial and extraaxial images) may be calculated very simply by means of a set of universal coefficients [P(i,j), Q(i,j), and R(i,j)] once the wave aberrations of selected rays are known. The values of these coefficients are uniquely determined by the form of polynomial assumed for the wave aberration and by the pattern of rays traced. Tables of P(i,j), Q(i,j), and R(i,j) are presented for the different cases that can arise in practice.