Lecture 9: Optimization and Correction

Nonlinear Optimization

Solution by linear algebra system matrix A
cases depending on the number of n/m

m=n: exact

m>n: underdetermined
parameters more than targets-->solution not unique

m<n:overdetermined parameters less than targets-->best approximation solution

Iterative numerical solution
strategy for optimization

direction of improvement step

size of improvement step

Steepest Descent Method

Constraints on Optimization

Boundary conditions and constraints

type of constraints

Numerical realizations

Lagrange multiplier

Penalty function:outside the permitted domain

Barrier function: inside the permitted domain

Regular variable, soft-constraint

Effect of constraints

possible not to obtain the optimizedest solution

Optimization and Starting Point

1.The initial starting point determines the final result

2.First Paraxial layout

3.Pre-calculation

0th properties of the system: focal length, magnification, pupil size and location, size/length of the system, image location

system structure (independent of lens bendings and 3rd order correction)
symmetry, field flattening correction, achromatism, apochromatism, distortion-correction, anastigmatism

Lens bending with 3rd order lens contributions
spherical aberration, coma, astigmatism

Optimization Merit Function

Performance criteria

Ray aberrations

Spot diameter

Wavefront description by Zernike coefficient, rms value

Strehl ratio,PSF

Contrast values for selected spatial frequencies

uniformity of illumination

Global optimization

Escape Function: Adding penalty function

Saddel Point Method

Adding a meniscus lens with thickness zero