Networks of local minima in optical system optimization

Area of Science:

• Optical engineering
• Computational optics
• Optimization theory

Background:

• The merit function landscape in optical system design often presents complex, multi-modal characteristics.
• Understanding the connectivity and topology of local minima is crucial for effective global optimization.

Purpose of the Study:

• To investigate a novel network feature within the merit function landscape of optical systems.
• To analyze the topological properties of connected local minima and their implications for optimization.

Main Methods:

• Analysis of the merit function landscape under specific design conditions.
• Identification and characterization of local minima and saddle points.
• Topological analysis of the network formed by local minima, using the symmetric Cooke triplet as a model system.

Main Results:

• A surprising network structure emerges where local minima are interconnected when certain conditions are met.
• Each connection (link) between adjacent local minima is characterized by a saddle point with a Morse index of 1.
• The study visualizes this network for a symmetric Cooke triplet, demonstrating the practical application of the findings.

Conclusions:

• The interconnected network of local minima represents a significant topological feature in optical design merit functions.
• The presence of saddle points with Morse index 1 facilitates movement between local minima, influencing global search strategies.
• This network understanding provides new insights for developing more efficient global optimization algorithms in optical system design.