Three-dimensional array diffraction-limited foci from Greek ladders to generalized Fibonacci sequences.

Area of Science:

• Optics and Photonics
• Number Theory

Background:

• The Greek ladder is a mathematical method for approximating real numbers with rational numbers.
• This technique has connections to continued fractions and generalized Fibonacci sequences.

Purpose of the Study:

• To design novel nano-photonic devices capable of producing three-dimensional (3D) array foci.
• To demonstrate that the focusing properties of these devices can be controlled using mathematical characteristics derived from the Greek ladder method.

Main Methods:

• Utilized continued fraction theory and algebraic equations to analyze the Greek ladder method.
• Implemented proper switching and phase modulation techniques (binary, ternary, quaternary) in nano-photonic device design.
• Designed devices to produce diffraction-limited array foci at desired focal planes.

Main Results:

• Successfully designed various nano-photonic devices based on the Greek ladder mathematical framework.
• Achieved the creation of 3D array foci with controllable focusing properties.
• Demonstrated the ability to freely design or distribute diffraction-limited array foci across multiple focal planes.

Conclusions:

• The Greek ladder mathematical technique provides a robust framework for designing advanced nano-photonic devices.
• This approach enables precise control over light focusing, leading to customizable 3D focal arrays.
• The developed technology has significant potential for applications requiring tailored light manipulation.