Florian Bociort

Networks of local minima in optical system optimization

We discuss a surprising new feature of the merit function landscape in optical system design. When certain conditions are satisfied, the local minima form a network in which all nodes are connected. Each link between two neighboring minima contains a saddle point with a Morse index of 1. For a simple global optimization search (the symmetric Cooke triplet), the network of the corresponding set of local minima is presented.

Determination of wavefront structure for a Hartmann Wavefront Sensor using a phase-retrieval method

We apply a phase retrieval algorithm to the intensity pattern of a Hartmann wavefront sensor to measure with enhanced accuracy the phase structure of a Hartmann hole array. It is shown that the rms wavefront error achieved by phase reconstruction is one order of magnitude smaller than the one obtained from a typical centroid algorithm. Experimental results are consistent with a phase measurement performed independently using a Shack-Hartmann wavefront sensor.

Network structure of the set of local minima in optical system optimization

We discuss a surprising new feature of the merit function landscape in optical system design. When certain conditions are satisfied, the set of local minima forms a network in which all nodes are connected. Each link between two neighboring minima contains a special type of saddle point (more precisely, a saddle point having a Morse index. On this basis, a new global optimization method that takes advantage of this feature is proposed. The central component of the new method, the algorithm for saddle point detection, works in a parameter space of arbitrary dimensionality, and uses only the local optimization engine of the optical design program. For a simple global optimization search (the symmetric Cooke triplet) the network of the corresponding set of local minima is presented.

Chromatic paraxial aberration coefficients for radial gradient-index lenses

A simple derivation of analytic expressions for the chromatic paraxial aberration coefficients of radial gradient-index lenses is presented. By decomposing the transverse chromatic aberration vector of an arbitrary paraxial ray in contributions from refraction at the surfaces and from transfer through the inhomogeneous media of the system, remarkably short formulas for the contributions of transfer through the gradient medium to the axial and lateral color coefficients are obtained. In the thin-lens approximation these expressions lead to well-known results for the total chromatic aberrations of a radial gradient-index lens.

Instabilities and fractal basins of attraction in optical system optimization

Many optical design programs use various forms of the damped least-squares method for local optimization. In this paper, we show that damped least-squares algorithms, with maximized computational speed, can create sensitivity with respect to changes in initial conditions. In such cases, starting points, which are very close to each other, lead to different local minima after optimization. Computations of the fractal capacity dimension show that sets of these starting points, which lead to the same minimum (the basins of attraction for that minimum), have a fractal structure. Introducing more damping makes the optimization process stable.

Network search method in the design of extreme ultraviolet lithographic objectives

The merit function space of mirror system for extreme ultraviolet (EUV) lithography is studied. Local minima situated in the multidimensional optical merit function space are connected via links that contain saddle points and form a network. We present networks for EUV lithographic objective designs and discuss how these networks change when control parameters, such as aperture and field, are varied, and constraints are used to limit the variation domain of the variables. A good solution in a network, obtained with a limited number of variables, has been locally optimized with all variables to meet practical requirements.

Saddle-point construction in the design of lithographic objectives, part 1: method

The multidimensional merit function space of complex optical systems contains a large number of local minima. We illustrate a method to find new local minima by constructing saddle points, with examples of deep and extreme UV objectives. The central idea of the method is that, at certain positions in a system with N surfaces that is a local minimum, a thin meniscus lens or two mirror surfaces can be introduced to con- struct a system with N+2 surfaces that is a saddle point. When optimi- zation rolls down on the two sides of the saddle point, two minima are obtained. Often one of these two minima can also be reached from sev- eral other saddle points constructed in the same way. With saddle-point construction we can obtain new design shapes from existing ones in a simple, efficient, and systematic manner that is suitable for complex de- signs such as those for lithographic objectives.

Finding new local minima in lens design landscapes by constructing saddle points

Finding good new local minima in the merit function land- scape of optical system optimization is a difficult task, especially for com- plex design problems where many minima are present. Saddle-point construction SPC is a method that can facilitate this task. We prove that, if the dimensionality of the optimization problem is increased in a way that satisfies certain mathematical conditions the existence of two independent transformations that leave the merit function unchanged, then a local minimum is transformed into a saddle point. With SPC, lenses are inserted in an existing design in such a way that subsequent optimizations on both sides of the saddle point result in two different system shapes, giving the designer two choices for further design. We present a simple and efficient version of the SPC method. In spite of theoretical novelty, the practical implementation of the method is very simple. We discuss three simple examples that illustrate the essence of the method, which can be used in essentially the same way for arbitrary systems.

Looking for order in the optical design landscape

In present-day optical system design, it is tacitly assumed that local minima are points in the merit function landscape without relationships between them. We will show however that there is a certain degree of order in the design landscape and that this order is best observed when we change the dimensionality of the optimization problem and when we consider not only local minima, but saddle points as well. We have developed earlier a computational method for detecting saddle points numerically, and a method, then applicable only in a special case, for constructing saddle points by adding lenses to systems that are local minima. The saddle point construction method will be generalized here and we will show how, by performing a succession of one-dimensional calculations, many local minima of a given global search can be systematically obtained from the set of local minima corresponding to systems with fewer lenses. As a simple example, the results of the Cooke triplet global search will be analyzed. In this case, the vast majority of the saddle points found by our saddle point detection software can in fact be obtained in a much simpler way by saddle point construction, starting from doublet local minima.

Saddle-point construction in the design of lithographic objectives, part 2: application

Optical designers often insert or split lenses in existing de- signs. Here, we apply in the design of objectives for deep and extreme UV lithography an alternative method for adding new components that consists of constructing saddle points in the optical merit function land- scape and obtaining new local minima from them. The design examples show that this remarkably simple method can be easily integrated with traditional design techniques. The new method has significantly im- proved our design productivity in all cases in which we have applied it so far. High-quality designs of lithographic objectives are obtained with this method.