The evaluation of the variance of the wave-aberration difference function

Hopkins' treatment of tolerance theory shows that, in designing quality optical systems, we should aim at minimizing the variance K of the wave-aberration difference function. Since the value of K is essentially positive, a useful criterion for the whole field is the sum of the suitably weighted values of K for a typical set of image points, corresponding to a range of selected values of spatial frequencies s and azimuths psi. In this paper we show that the variance K (for both the axial and the extraaxial images) may be calculated by means of a set of universal coefficients P(i,j;s), Q(i,j;spsi) and R(i,j;spsi) once the wave-aberrations of certain rays are known. The values of these coefficients are uniquely determined by the function of the wave-aberration polynomial assumed and by the-pattern of rays traced. Examples of the coefficients are presented.