Variational approach to calculation of light field eikonal function for illuminating a prescribed region

Area of Science:

• Optics
• Mathematical Physics

Background:

• Calculating the light field eikonal function is crucial for precise light focusing.
• Existing methods can be computationally intensive.

Purpose of the Study:

• To develop an efficient method for calculating the light field eikonal function for focused light.
• To connect eikonal function calculation with mass transportation problems.

Main Methods:

• Formulating the eikonal function calculation as a variational problem.
• Applying the Monge-Kantorovich mass transportation problem framework.
• Reducing the problem to linear programming and linear assignment problems.

Main Results:

• The cost function in the mass transportation problem directly relates to the distance between source and target regions.
• The eikonal function corresponds to a mapping that minimizes total distance.
• The method simplifies eikonal function calculation to linear programming.

Conclusions:

• The proposed mass transportation approach provides an efficient and mathematically rigorous method for calculating the eikonal function.
• This formalism simplifies complex optical focusing problems into solvable linear programming tasks.
• The method is demonstrated with practical examples of focusing light into specific shapes.