Damped least squares method using diagonal elements of second order derivative as a damping term is studied. The results are compared with the those obtained from other methods which use the first order derivative. In this study, the methods are applied to the two different optical systems, the Cassegrain-inverse Cassegrain type mirror system and the double-Gauss type camera lens system.
In the mirror system design, both the least squares method and the damped least squares method are used for the purpose of comparison. Three kinds of damping forms are used in the damped least squares method, and these damping forms are a) additive damping, b) multiplicative damping and c) diagonal elements of second derivative damping. The damping factors for the damped least squares method are chosen 10 different values from the range of 0.0007 to 10.
When the mirror system is optimized by 4 variables, (2 curvatures and 2 conic constants), the merit function of optical system converges to two minima. One is 15.30% and the other is 14.5% with respect to that of initial design. And when the mirror system is optimized by 10 variables (4 curvatures, 4 inter-surface distances, 2 conic constants), the merit function of optical system converges to a minimum which is 1.47% with respect to that of initial design. In the mirror system design, the converging rate of diagonal elements of second order derivative damping is found faster than other methods. When the stability is concerned, the diagonal elements of second order derivative damping is more stable than others.
In the camera lens system design, only damped least squares method is used because of instability of least squares method. The damping factors are used the same values with the case of mirror system design. When the camera lens system is optimized by 10 curvature variables, the merit function of optical system converges to a minimum which is 4.2% with respect to that of initial design. The converging rate of diagonal elements of second order derivative damping is faster than others. When the stability is concerned, diagonal elements of second order derivative damping is found more stable than others.
In conclusion, it is found that the damped least squares method with the diagonal elements of second order derivative damping gives generally improved convergence and stability.
