Automatic Correction of First and Third Order Aberrations

Abstract

The problem of automatic lens design is generally shown to be equivalent to minimize a merit function Φ (x₁••••x_n) whose magnitude depends upon the residual aberrations of the system. In this report, the merit function Φ is computed at n-dimensional lattice points (μ₁, μ₂ …… μ_n) distributed over the region of independent variables where a set of positive integers (μ₁, μ₂ •••• μ_n) represents each lattice point.
Φ (μ₁•••• μ_n) can be expanded to a series of “sub-functions” Φ_k (μ₁, μ₂••••μ_k, μ_k+1.0••••μ_n0) each of which is the intersection of Φ with the k-dimensional sub-space containing a starting point μ₀(μ₁₀, μ₂₀••••μ_n0).
When the minimum of the sub-function Φ_k is known, the procedure of finding the minimum of Φ_k+1 is derived and this procedure of “sub-minimization” is applied successively to a series of sub-function Φ₁, Φ₂•••• until the final minimum of Φ_n(=Φ) in the n-dimensional space is reached. (successive minimization method)
Both the extremely slow convergence and complicated numerical computations of the steepest descent method are avoided by the present method.
In case of repeated applications of the successive minimization method corresponding to the continuous variation of external parameters included in Φ, both the starting point and the lattice distance are improved at each minimization process, the improvement being based upon the results of the previous minimization. (variable metric method) By this technique, the accuracy of solutions is remarkably increased.
The whole methods described above have definite advantages when applied to the problem of automatic correction of first- and third-order aberrations. In this case, the numerical computing process becomes considerably simple by means of lattice-points expression of the present method.
As numerical examples, the whole computing processes of triplet design are presented. The variation of residual third-order spherical aberration, when all the other first- and third-order aberrations are corrected to target values, is also computed against various values of thin-lens system parameters.