Focused fields of given power with maximum longitudinal electric field component inside a substrate.

Area of Science:

• Optics and Photonics
• Electromagnetism

Background:

• Focusing light through interfaces is crucial in various optical applications.
• Maximizing the longitudinal electric field component is key for certain high-intensity applications.
• Traditional methods often use annular illumination, which may not be optimal.

Purpose of the Study:

• To derive closed-form formulas for the electromagnetic field in a lens pupil.
• To determine the pupil field that maximizes the longitudinal electric field upon focusing through an interface at any depth.
• To explore alternative pupil field distributions beyond annular illumination.

Main Methods:

• Derivation of closed-form expressions for electromagnetic fields.
• Optimization of pupil field functions to maximize a specific field component.
• Numerical simulations and case studies for various parameters.

Main Results:

• Closed formulas for electromagnetic fields were obtained.
• The optimal pupil field is a continuous, monotonically increasing function of the radial coordinate.
• This optimal field differs significantly from conventional annular illumination.
• The study analyzed various scenarios including different materials, numerical apertures (NA), and focusing depths.
• The influence of absorbing media was also investigated.

Conclusions:

• A novel pupil field distribution is proposed for enhanced focusing.
• The findings offer a new approach for optimizing light focusing through interfaces.
• The presented formulas and results are valuable for designing advanced optical systems.