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.

Abstract. 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 construct a system with N + 2 surfaces that is a saddle point. When optimization 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 several 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 designs such as those for lithographic objectives.

The multidimensional merit function space of complex optical systems contains a large number of local minima that are connected via links that contain saddle points. In this work, we illustrate a method to construct such saddle points with examples of deep UV objectives and extreme UV mirror systems for lithography. The central idea of our 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 construct a system with N+2 surfaces that is a saddle point. When the optimization goes down on the two sides of the saddle point, two minima are obtained. We show that often one of these two minima can be reached from several other saddle points constructed in the same way. The practical advantage of saddle-point construction is that we can produce new designs from the existing ones in a simple, efficient and systematic manner.Keywords: saddle point, lithography, optimization, optical system design, EUV CONSTRUCTING SADDLE POINTS IN THE MERIT FUNCTION LANDSCAPEIn optical system design the multidimensional merit function space typically comprises a large number of local minima. It has been shown recently 1 that these local minima are connected via optimization paths that start from a specific type of saddle point (saddle point with Morse index of 1) and form a network. For complex systems the detection of the entire network is difficult and time consuming. In an accompanying paper in this volume, an efficient and fast method to construct saddle points with Morse index of one is described 2 . This method is illustrated in the present article with examples of lithographic objectives for deep and extreme UV.A point in the merit function space for which the gradient of the merit function (MF) vanishes is called a critical point. At critical points, for which the Hessian matrix of the second-order derivatives of the MF with respect to the optimization variables has a non-zero determinant, the number of negative eigenvalues of the Hessian gives the socalled Morse index (MI). A negative eigenvalue means that along the corresponding eigenvector the critical point is a maximum and a positive eigenvalue means a minimum along the corresponding direction. Local minima have MI = 0, local maxima have MI = N and saddle points have MI between 1 and N-1. For the network structure it is sufficient to consider saddle points with MI = 1, i.e. saddle points that are maxima along one direction. From such a saddle point, two distinct local minima can be generated by letting the optimization go down on its two sides along that direction. The optimization paths, together with the saddle point with MI = 1, form a link in the optimization space between the two minima.From a given local minimum with N surfaces we can construct saddle points with MI = 1 having N+2 surfaces by inserting at any surface in the local minimum a zero-thickness meniscus lens (or two mirror surfaces with zero distances between them) 2 . (See Fig. 1.) In this paper, we a...

The merit function space of mirror systems for EUV lithography is studied. Local minima situated in a multidimensional merit function space are connected via links that contain saddle points and form a network. In this work we present the first networks for EUV lithographic objectives 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 (SPC) is a new method to insert lenses into an existing design. With SPC, by inserting and extracting lenses new system shapes can be obtained very rapidly, and we believe that, if added to the optical designer's arsenal, this new tool can significantly increase design productivity in certain situations. Despite the fact that the theory behind SPC contains mathematical concepts that are still unfamiliar to many optical designers, the practical implementation of the method is actually very easy and the method can be fully integrated with all other traditional design tools. In this work we will illustrate the use of SPC with examples that are very simple and illustrate the essence of the method. The method can be used essentially in the same way even for very complex systems with a large number of variables, in situations where other methods for obtaining new system shapes do not work so well.